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Right triangle

A right triangle is a triangle in which one interior angle measures exactly 90 degrees.^[1] The side opposite this right angle, known as the hypotenuse, is the longest side of the triangle.^[2] The two sides that form the right angle are called the legs.^[3] The defining property of right triangles is encapsulated in the Pythagorean theorem, which states that the square of the hypotenuse's length equals the sum of the squares of the legs' lengths: a^2 + b^2 = c^2, where c is the hypotenuse.^[4] This relation holds exclusively for right triangles and enables the calculation of unknown side lengths when two sides and the right angle are known.^[5] The theorem's proof and applications extend to fields like physics and engineering, underscoring the triangle's foundational role in mathematics.^[6] In right triangles, the two non-right angles are acute and complementary, summing to 90 degrees.^[7] Trigonometric functions—sine, cosine, and tangent—are defined using side ratios relative to these acute angles: for an angle \theta, \sin \theta = opposite/hypotenuse, \cos \theta = adjacent/hypotenuse, and \tan \theta = opposite/adjacent.^[8] These ratios form the basis of trigonometry and are essential for solving problems involving distances, heights, and angles in real-world scenarios such as surveying and navigation.^[1] Special cases of right triangles include the isosceles right triangle (45-45-90), where the legs are equal and the hypotenuse is \sqrt{2} times a leg's length, and the 30-60-90 triangle, with side ratios of $1 : \sqrt{3} : 2.^[9] These configurations arise frequently in geometric constructions and derivations, highlighting the versatility of right triangles in both theoretical and applied contexts.^[10]

Definition and Characterization

Definition

A triangle is a three-sided polygon formed by three line segments connecting three non-collinear points, known as vertices, with the segments called sides and the points where they meet called angles.^[11] In Euclidean geometry, the sum of the interior angles of any triangle is exactly 180 degrees.^[12] A right triangle, also known as a right-angled triangle, is a specific type of triangle that contains exactly one interior angle measuring 90 degrees, or a right angle.^[13] This distinguishes it from an acute triangle, where all three angles are less than 90 degrees, and an obtuse triangle, which has one angle greater than 90 degrees and two less than 90 degrees.^[14] The right angle is typically denoted as angle C, with the two sides forming this angle referred to as the legs, labeled a and b, while the side opposite the right angle is the hypotenuse, labeled c, which is the longest side.^[13] The concept of the right triangle has been recognized since ancient times within the framework of Euclidean geometry, as formalized in Euclid's Elements around 300 BCE, where right angles and their properties in triangles are defined and explored as foundational elements of plane geometry.^[15] This definition implies certain characteristic relations among the sides, such as those encapsulated in the Pythagorean theorem.^[13]

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle with legs a and b and hypotenuse c, the square of the hypotenuse equals the sum of the squares of the legs:

c^2 = a^2 + b^2.

^[16] This relation, formalized in Euclid's Elements (Book I, Proposition 47), holds for any right-angled triangle where the right angle is between the legs a and b.^[17] One simple proof uses similar triangles. Consider a right triangle ABC with right angle at C, legs a = BC and b = AC, and hypotenuse c = AB. Drop an altitude from C to the hypotenuse AB, meeting at point D. This creates two smaller right triangles ACD and BCD, both similar to the original triangle ABC. The similarity gives the ratios \frac{a}{c} = \frac{p}{a} where p = AD, and \frac{b}{c} = \frac{q}{b} where q = DB (with p + q = c). Rearranging yields a^2 = c p and b^2 = c q, so a^2 + b^2 = c(p + q) = c^2.^[18] The converse of the theorem also holds: if in any triangle with sides a, b, and c (where c is the longest side) the relation c^2 = a^2 + b^2 is satisfied, then the triangle is a right triangle with the right angle opposite side c.^[19] This is proven in Euclid's Elements (Book I, Proposition 48) by assuming the relation and showing congruence of triangles that implies a right angle.^[20] Integer solutions to the theorem, known as Pythagorean triples, include primitive triples where \gcd(a, b, c) = 1. These can be generated using the formula a = m^2 - n^2, b = 2mn, c = m^2 + n^2, where m > n > 0 are coprime integers not both odd.^[21] For example, with m=2, n=1, the triple is (3, 4, 5).^[22] A basic application is verifying whether a triangle with measured sides forms a right angle by checking if the sum of the squares of two sides equals the square of the third.^[23]

Thales' Theorem

Thales' theorem states that if A and B are the endpoints of a diameter of a circle, and C is any other point on the circumference, then \angle ACB measures 90 degrees.^[24] This result characterizes right triangles through their relationship to circle geometry, providing an equivalent condition to the standard definition based on angles.^[25] The theorem is attributed to Thales of Miletus, a pre-Socratic philosopher active around 600 BCE, and is considered one of the earliest recorded theorems in Greek geometry.^[26] Ancient sources, including Eudemus's history of geometry as preserved by Proclus, credit Thales with this insight, which he reportedly used in practical measurements, such as determining distances at sea.^[27] A standard proof relies on the inscribed angle theorem, which establishes that an angle inscribed in a semicircle subtends 90 degrees. Consider the circle with center O and diameter AB. Since OA = OB = OC (all radii), triangles AOC and BOC are isosceles. The base angles in \triangle AOC are equal, so \angle OAC = \angle OCA, and their sum with \angle AOC = 180^\circ implies each base angle is (180^\circ - \angle AOC)/2. Similarly for \triangle BOC. Adding these, \angle ACB = 180^\circ - \angle AOB. But \angle AOB, the central angle subtending arc AB (a semicircle), is 180^\circ, so \angle ACB = 90^\circ.^[28] The converse of Thales' theorem holds: if \triangle ABC is right-angled at C, then the hypotenuse AB serves as the diameter of the circumcircle passing through A, B, and C.^[29] This follows by reversing the construction: the circle with diameter AB will have C on its circumference precisely because \angle ACB = 90^\circ.^[30] Geometrically, the theorem enables a simple construction to verify or create a right angle: draw a circle with the intended hypotenuse as diameter, then select any point C on the circumference to form \triangle ABC with right angle at C.^[31]

Sides and Angles

Side Relations

In a right triangle, the sides consist of two legs, denoted as a and b, and the hypotenuse c opposite the right angle, satisfying c = \sqrt{a^2 + b^2}. The hypotenuse is the longest side, as c^2 = a^2 + b^2 > a^2 implies c > a, and similarly c > b.^[32] The semiperimeter s of a right triangle is given by s = (a + b + c)/2. This quantity appears in geometric inequalities, including the isoperimetric inequality for triangles, which bounds the area relative to the perimeter and aids in distinguishing triangle types based on side lengths and perimeter efficiency.^[33] A triangle with side lengths a, b, and c (where c is the longest) is a right triangle if and only if a^2 + b^2 = c^2, providing a side-length characterization equivalent to the Pythagorean theorem. Additionally, the altitude from the right angle to the hypotenuse divides the original triangle into two smaller right triangles, each similar to the original and to each other, due to corresponding angles being equal.^[18] Right triangles with integer side lengths form Pythagorean triples (a, b, c) satisfying a^2 + b^2 = c^2. A primitive Pythagorean triple has \gcd(a, b, c) = 1, and any non-primitive integer triple is a positive integer multiple k(a, b, c) of a primitive one, where k > 1.^[34]

Angle Properties

In a right triangle, exactly one angle measures 90 degrees and is designated as the right angle. The remaining two angles, denoted as α and β, are acute angles, each measuring less than 90 degrees, and their measures sum to 90 degrees, rendering them complementary.^[9] This complementary relationship arises directly from the fact that the interior angles of any triangle sum to 180 degrees.^[7] Consequently, a right triangle cannot contain an obtuse angle (greater than 90 degrees), as including such an angle alongside the 90-degree angle would exceed the total angle sum.^[35] The acute angles α and β are positioned such that each is formed between the hypotenuse and one of the two legs. Specifically, for angle α, one leg lies opposite to it, while the other leg is adjacent to it; the hypotenuse remains opposite the right angle. These positional relations to the legs uniquely define the acute angles in terms of the triangle's sides.^[36] The presence of a single 90-degree angle serves as the primary characterization of a right triangle. Additional angle-based tests can verify this property without direct measurement of side lengths. For example, if two angles α and β in a triangle satisfy \tan \alpha \cdot \tan \beta = 1, then \alpha + \beta = 90^\circ due to the tangent addition formula, implying the third angle is 90 degrees.^[37] Another such test uses the identity \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 + 2 \cos \alpha \cos \beta \cos \gamma for any triangle; the sum equals 2 precisely when one cosine is zero (i.e., one angle is 90 degrees), confirming the right triangle property.^[38]

Area and Perimeter

Area Formulas

The area of a right triangle, where the legs are the two sides forming the right angle and denoted as a and b, is computed using the formula

A = \frac{1}{2} a b.

This expression arises directly from the base-height principle of triangle areas, as the perpendicular nature of the right angle positions leg b as the height relative to base a, or vice versa.^[39]^[40] More formally, the formula derives from the general area expression for any triangle with sides a and b enclosing angle C:

A = \frac{1}{2} a b \sin C.

For a right triangle, the included angle C = 90^\circ, and since \sin 90^\circ = 1, the expression simplifies to \frac{1}{2} a b. This contrasts with non-right triangles, where \sin C < 1 requires computing the sine value, complicating the area determination without additional measurements.^[41]^[39] An equivalent formulation uses one leg as the base and the altitude to that leg: A = \frac{1}{2} \times a \times h_a, where h_a is the altitude from the opposite vertex to side a. In a right triangle, h_a = b because the right angle ensures perpendicularity, reducing this to the primary formula; the same holds when interchanging a and b.^[39]

Semiperimeter and Heron's Formula

The semiperimeter s of a right triangle with legs a and b, and hypotenuse c, is defined as s = \frac{a + b + c}{2}.^[42] This quantity represents half the perimeter and plays a central role in Heron's formula, a general expression for the area K of any triangle in terms of its side lengths: K = \sqrt{s(s - a)(s - b)(s - c)}.^[42] Developed by Heron of Alexandria around the 1st century AD, this formula allows area computation without direct measurement of heights or angles.^[42] In the case of a right triangle, Heron's formula simplifies to the standard area expression K = \frac{1}{2}ab, demonstrating equivalence between the general method and the right-triangle-specific approach.^[43] Substituting c = \sqrt{a^2 + b^2} from the Pythagorean theorem yields s - c = \frac{a + b - c}{2}, with similar expressions for s - a and s - b. The product s(s - a)(s - b)(s - c) then algebraically expands to \left( \frac{1}{2}ab \right)^2, confirming that \sqrt{s(s - a)(s - b)(s - c)} = \frac{1}{2}ab.^[43] This simplification provides a verification tool for right-triangle properties. Specifically, applying Heron's formula to a triangle with sides a, b, and c and obtaining an area of \frac{1}{2}ab indicates the presence of a right angle between sides a and b, as the equality holds precisely when c = \sqrt{a^2 + b^2}.^[44]

Altitudes and Medians

Altitudes

In a right triangle with legs of lengths a and b, and hypotenuse of length c, the altitudes to the legs are straightforward due to the right angle. The altitude from the vertex opposite leg a to side a is simply h_a = b, as the perpendicular distance aligns with the other leg perpendicular to the base. Similarly, the altitude to leg b is h_b = a.^[13] These relations follow from the general formula for the altitude to a side, h = \frac{2 \times \text{area}}{\text{side length}}, where the area of the right triangle is \frac{1}{2}ab. The altitude to the hypotenuse, h_c, is more involved and plays a key role in the triangle's geometric properties. It is given by h_c = \frac{ab}{c}, derived by equating the area expressions: \frac{1}{2}ab = \frac{1}{2}c h_c. This altitude, drawn from the right-angled vertex perpendicular to the hypotenuse, intersects the hypotenuse at a point that divides it into two segments, denoted p and q, where p + q = c. The lengths of these segments are p = \frac{a^2}{c} (the segment adjacent to leg a) and q = \frac{b^2}{c} (adjacent to leg b).^[13]^[45] A significant property arises from the similarity of the two smaller right triangles formed by this altitude to the original triangle and to each other. Each leg serves as the geometric mean of the hypotenuse and the adjacent segment: a^2 = c p and b^2 = c q. Additionally, the altitude itself is the geometric mean of the two segments: h_c^2 = p q. These relations, known as the geometric mean theorems for right triangles, highlight the altitude's role in creating proportional similarities.^[45]^[13]

Medians

In a right triangle with legs of lengths a and b and hypotenuse of length c, the medians are line segments connecting each vertex to the midpoint of the opposite side. The lengths of these medians can be derived using Apollonius's theorem, which relates the length of a median to the side lengths of any triangle: for the median m_a to side a, m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}.^[46] Applying this to the median from the vertex opposite leg a (i.e., the median to side a), the formula simplifies due to the Pythagorean theorem a^2 + b^2 = c^2:

m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} = \frac{1}{2} \sqrt{2b^2 + 2(a^2 + b^2) - a^2} = \frac{1}{2} \sqrt{a^2 + 4b^2} = \frac{\sqrt{a^2 + 4b^2}}{2}.

Similarly, the median to leg b is m_b = \frac{\sqrt{b^2 + 4a^2}}{2}.^[46] The median to the hypotenuse, drawn from the right-angled vertex to the midpoint of side c, has a distinctive property: its length is exactly half the hypotenuse, so m_c = \frac{c}{2}. This follows from Apollonius's theorem as well:

m_c = \frac{1}{2} \sqrt{2a^2 + 2b^2 - c^2} = \frac{1}{2} \sqrt{2(a^2 + b^2) - c^2} = \frac{1}{2} \sqrt{2c^2 - c^2} = \frac{1}{2} \sqrt{c^2} = \frac{c}{2}.

Geometrically, this occurs because the midpoint of the hypotenuse is the circumcenter of the right triangle, and the circumradius R equals c/2; thus, the distance from the right-angled vertex to this midpoint is R.^[47]^[48] A general relation among the medians, applicable to right triangles, states that the sum of the squares of the medians equals \frac{3}{4} times the sum of the squares of the sides: m_a^2 + m_b^2 + m_c^2 = \frac{3}{4}(a^2 + b^2 + c^2). Substituting the right triangle medians yields \left(\frac{a^2 + 4b^2}{4}\right) + \left(\frac{b^2 + 4a^2}{4}\right) + \left(\frac{c^2}{4}\right) = \frac{3}{4}(a^2 + b^2 + c^2), confirming the property through the symmetries of the Pythagorean relation.

Inradius and Incircle

In a right triangle with legs of lengths a and b and hypotenuse of length c, the inradius r is the radius of the incircle, which is tangent to all three sides. The formula for the inradius is r = \frac{a + b - c}{2}.^[49] This expression arises from the general relation r = \frac{A}{s}, where A is the area and s = \frac{a + b + c}{2} is the semiperimeter; substituting A = \frac{ab}{2} yields r = \frac{ab}{a + b + c}, which simplifies to the given form using properties specific to right triangles.^[50] The points of tangency of the incircle provide a geometric interpretation. The circle touches each leg at a distance r from the right-angled vertex, due to the equal tangent segments from that vertex and the 90-degree angle aligning the incenter at coordinates (r, r) when the vertex is at the origin with legs along the axes. On the hypotenuse, the points of tangency divide it into segments of lengths a - r and b - r from the acute vertices, so c = (a - r) + (b - r) = a + b - 2r, confirming r = \frac{a + b - c}{2}.^[49] The fundamental relation between area, inradius, and semiperimeter, A = r s, applies to right triangles and verifies the formula: \frac{ab}{2} = r \cdot \frac{a + b + c}{2}.^[51] This equality holds because the incircle divides the triangle into three smaller triangles of equal height r, with bases summing to the perimeter. A characterizing property is that if a triangle with sides a, b, c (where c is the longest side) has inradius r = \frac{a + b - c}{2} = s - c, then the angle opposite c must be 90 degrees. This follows from the general half-angle formula \tan\left(\frac{C}{2}\right) = \frac{r}{s - c}, which implies \tan\left(\frac{C}{2}\right) = 1, so \frac{C}{2} = 45^\circ and C = 90^\circ.^[52]

Circumradius and Circumcircle

In a right triangle, the circumcircle is the unique circle passing through all three vertices, with the hypotenuse serving as its diameter according to Thales' theorem.^[53] This theorem states that if a triangle is inscribed in a circle where one side is the diameter, the opposite angle is a right angle, and conversely, for any right triangle, the hypotenuse is the diameter of the circumcircle.^[53] The center of the circumcircle, or circumcenter, is the midpoint of the hypotenuse./02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles) Thus, the circumradius R, defined as the radius of this circle, equals half the hypotenuse length: R = \frac{c}{2}, where c denotes the hypotenuse./02%3A_General_Triangles/2.05%3A_Circumscribed_and_Inscribed_Circles) This result aligns with the general circumradius formula for any triangle, R = \frac{abc}{4K}, where a and b are the legs, c the hypotenuse, and K the area.^[54] For a right triangle, K = \frac{1}{2}ab, so

R = \frac{abc}{4 \cdot \frac{1}{2}ab} = \frac{abc}{2ab} = \frac{c}{2}.

^[54] A key property is that the right-angled vertex subtends the diameter, ensuring the angle at that vertex is 90 degrees by the inscribed angle theorem.^[53]

Exradii

In a right triangle with the right angle at vertex C and legs a and b adjacent to the acute angles A and B, respectively, and hypotenuse c opposite C, the exradii are the radii of the excircles, each tangent to one side of the triangle and to the extensions of the other two sides.^[55] The excircle opposite angle A touches side BC (of length a) internally and the extensions of sides AB (length c) and AC (length b) externally, while similar tangency configurations apply to the excircles opposite B and C.^[55] The exradius r_a opposite angle A is given by the general formula r_a = \Delta / (s - a), where \Delta = ab/2 is the area and s = (a + b + c)/2 is the semiperimeter; this simplifies to r_a = s - b = (a + c - b)/2 for a right triangle.^[56]^[57] Similarly, the exradius r_b opposite B is r_b = s - a = (b + c - a)/2, and the exradius r_c opposite the right angle C is r_c = s = (a + b + c)/2.^[57] The sum of the exradii satisfies the relation r_a + r_b + r_c = 4R + r, where R = c/2 is the circumradius and r = (a + b - c)/2 is the inradius; substituting these yields r_a + r_b + r_c = (a + b + 3c)/2, which aligns with the direct sum (s - b) + (s - a) + s = s + c.^[58]^[56] These explicit forms highlight the exradii's dependence on the side lengths, with r_c notably equaling the semiperimeter due to the right angle at C.^[57]

Trigonometric Ratios

Primary Ratios

In a right triangle, the primary trigonometric ratios—sine, cosine, and tangent—are defined in terms of the sides relative to one of the acute angles, denoted as \alpha. These ratios relate the angle to the lengths of the opposite side (a), adjacent side (b), and hypotenuse (c).^[59] The sine of angle \alpha is the ratio of the length of the side opposite \alpha to the hypotenuse:

\sin \alpha = \frac{a}{c}

The cosine of angle \alpha is the ratio of the length of the side adjacent to \alpha (excluding the hypotenuse) to the hypotenuse:

\cos \alpha = \frac{b}{c}

This ratio complements the sine, reflecting the projection of the adjacent side along the hypotenuse, a concept central to vector decompositions in classical mechanics and geometry.^[60] The tangent of angle \alpha is the ratio of the length of the side opposite \alpha to the adjacent side:

\tan \alpha = \frac{a}{b}

Equivalently, \tan \alpha = \frac{\sin \alpha}{\cos \alpha}, this ratio is fundamental in applications such as slope calculations in coordinate geometry and surveying. For the right angle itself, denoted as $90^\circ, the trigonometric ratios take special values: \sin 90^\circ = 1, since the opposite side equals the hypotenuse; \cos 90^\circ = 0, as the adjacent side has zero length relative to the hypotenuse; and \tan 90^\circ is undefined, due to division by zero. These values follow directly from the limiting behavior of the ratios as the angle approaches $90^\circ. In a right triangle, the two acute angles \alpha and \beta are complementary, satisfying \alpha + \beta = 90^\circ. Consequently, \sin \alpha = \cos \beta, \cos \alpha = \sin \beta, and \tan \alpha = \cot \beta (where \cot is the reciprocal of tangent). This cofunction property simplifies computations and underpins many trigonometric identities.

Reciprocal and Quotient Ratios

In a right triangle, the reciprocal trigonometric ratios—cosecant, secant, and cotangent—provide inverse measures relative to the primary ratios of sine, cosine, and tangent, respectively, expressed directly in terms of the triangle's side lengths. These functions are particularly useful for acute angles, where they relate the hypotenuse or adjacent side to the opposite side. The cosecant of an acute angle \alpha is defined as the reciprocal of the sine:

\csc \alpha = \frac{1}{\sin \alpha} = \frac{c}{a},

where c is the hypotenuse and a is the side opposite \alpha.^[61]^[62] The secant of \alpha is the reciprocal of the cosine:

\sec \alpha = \frac{1}{\cos \alpha} = \frac{c}{b},

with b denoting the side adjacent to \alpha.^[61]^[62] The cotangent of \alpha is the reciprocal of the tangent:

\cot \alpha = \frac{1}{\tan \alpha} = \frac{b}{a}.

This ratio emphasizes the adjacency-to-opposition relationship without involving the hypotenuse.^[61]^[62] These reciprocal ratios apply specifically to the acute angles in a right triangle, as the right angle of $90^\circ renders the secant undefined (since \cos 90^\circ = 0), the cosecant equals 1 (since \sin 90^\circ = 1), and the cotangent equals 0. The quotient aspects arise in the relations between primary ratios, such as \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{a}{b}, with its reciprocal \cot \alpha = \frac{\cos \alpha}{\sin \alpha} = \frac{b}{a} mirroring the side-based quotient.^[61]^[62]

Special Right Triangles

45-45-90 Triangle

A 45-45-90 triangle is an isosceles right triangle characterized by two equal acute angles measuring 45 degrees each and a right angle of 90 degrees.^[4] This configuration arises naturally when the diagonal of a square is drawn, dividing the square into two congruent 45-45-90 triangles.^[63] In such a triangle, the two legs are of equal length, denoted as a, while the hypotenuse measures a\sqrt{2}, following the side ratio of $1:1:\sqrt{2}.^[64] This hypotenuse length derives from the Pythagorean theorem applied to the legs: a^2 + a^2 = c^2, yielding c = a\sqrt{2}.^[4] The area of the triangle is given by \frac{1}{2}a^2, reflecting the standard formula for right triangles with equal legs.^[4] The trigonometric ratios for the 45-degree angles are symmetric due to the isosceles nature: \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} and \tan 45^\circ = 1.^[4] These values, often rationalized as \frac{1}{\sqrt{2}} for sine and cosine, provide exact computations in problems involving these angles.^[64] Applications of the 45-45-90 triangle include determining diagonals in square-based structures, such as floor tiles or screen dimensions, where the diagonal length is the leg times \sqrt{2}.^[63] It also features in basic geometric constructions, like bisecting angles or creating perpendicular lines using a compass and straightedge.^[65]

30-60-90 Triangle

A 30-60-90 triangle is a right triangle with acute angles measuring 30° and 60°.^[66] The side lengths follow a fixed ratio: the side opposite the 30° angle (short leg) is x, the side opposite the 60° angle (long leg) is x\sqrt{3}, and the hypotenuse is $2x, for some positive real number x.^[66] This ratio can be derived by bisecting an equilateral triangle, which has all sides equal and all angles 60°. Drawing an altitude from one vertex to the base splits the equilateral triangle into two congruent 30-60-90 triangles, where the altitude forms the long leg, half the base is the short leg, and the original side is the hypotenuse.^[67] Applying the Pythagorean theorem confirms the relation: (x)^2 + (x\sqrt{3})^2 = (2x)^2, simplifying to x^2 + 3x^2 = 4x^2.^[67] The trigonometric ratios for these angles are standard values derived from the side ratios. For the 30° angle: \sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}, \tan 30^\circ = \frac{1}{\sqrt{3}}. For the 60° angle: \sin 60^\circ = \frac{\sqrt{3}}{2}, \cos 60^\circ = \frac{1}{2}, \tan 60^\circ = \sqrt{3}.^[68]^[69] The area of a 30-60-90 triangle with short leg x is \frac{\sqrt{3}}{2} x^2, computed as half the product of the legs.^[66] Equivalently, with hypotenuse a = 2x, the area is \frac{\sqrt{3}}{8} a^2.^[66] In applications, the 30-60-90 triangle arises when finding the height of an equilateral triangle, as the altitude divides it into two such right triangles, allowing computation of the height as \frac{\sqrt{3}}{2} times the side length.^[70]

Kepler Triangle

The Kepler triangle is a special right triangle whose side lengths form a geometric progression with common ratio the square root of the golden ratio, \phi = \frac{1 + \sqrt{5}}{2} \approx 1.61803. It is named for the German mathematician and astronomer Johannes Kepler (1571–1630), who first described the triangle in a letter to his former teacher Michael Mästlin dated October 30, 1597, noting its connection to the golden ratio in the context of geometric proportions. Although Kepler did not name the triangle himself—the term "Kepler triangle" was introduced later by mathematician Roger Herz-Fischler in 1979—the shape has since been recognized for embedding the golden ratio in a Pythagorean framework.^[71]^[72] The side lengths of the Kepler triangle are in the ratio $1 : \sqrt{\phi} : \phi, where the two legs are of lengths 1 and \sqrt{\phi} \approx 1.27202 (with the shorter leg opposite the smaller acute angle), and the hypotenuse is \phi. This configuration satisfies the Pythagorean theorem, as $1^2 + (\sqrt{\phi})^2 = 1 + \phi = \phi^2, leveraging the defining property of the golden ratio \phi^2 = \phi + 1. The acute angles are approximately $38.173^\circ (opposite the shorter leg) and $51.827^\circ (opposite the longer leg), with the right angle between the legs; these irrational measures distinguish the Kepler triangle from other special right triangles with rational-degree angles.^[72]^[73] Key properties of the Kepler triangle include its area and perimeter expressed in terms of \phi: assuming the shorter leg is 1, the area is \frac{1}{2} \sqrt{\phi} \approx 0.63604, and the perimeter is $1 + \sqrt{\phi} + \phi \approx 4.89005. These relations highlight the triangle's efficiency in geometric constructions involving the golden ratio, which governs the proportions of the regular pentagon—for instance, the ratio of a pentagon's diagonal to its side is \phi, and the Kepler triangle can be derived from subdividing pentagonal elements, linking it directly to pentagon geometry and symmetry.^[72]^[74] The trigonometric values of the acute angles further tie the Kepler triangle to broader mathematical phenomena, such as the logarithmic spiral. For the larger acute angle \alpha \approx 51.827^\circ, \tan \alpha = \sqrt{\phi} \approx 1.27202, but the structure's golden ratio progression facilitates constructions approximating spirals where the growth factor aligns with \phi, as in the golden spiral—a logarithmic spiral that expands by \phi every quarter turn, often visualized through sequences of golden-ratio-based right triangles like the Kepler.^[72]^[75]

Advanced Properties

Euler Line

In a right triangle, the Euler line is the straight line that passes through the orthocenter, located at the vertex of the right angle; the centroid; and the circumcenter, which lies at the midpoint of the hypotenuse.^[76]^[77] This configuration aligns several key triangle centers along a single path originating from the right-angled vertex.^[78] The centroid divides the segment connecting the orthocenter and circumcenter in the ratio 2:1, with the longer portion from the orthocenter to the centroid.^[76] In this setup, the Euler line coincides with the median drawn from the right-angled vertex to the midpoint of the hypotenuse.^[79] This median serves as the direct path linking these centers, simplifying the geometric alignment compared to non-right triangles.^[80] The nine-point circle, which passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices, has its center at the midpoint of the orthocenter-circumcenter segment and a radius equal to half the circumradius R/2.^[81] This center lies on the Euler line, further integrating the circle's properties with the line's structure.^[82] The placement of the orthocenter at a vertex in a right triangle results in a more straightforward Euler line than in acute or obtuse triangles, where the orthocenter is interior or exterior, respectively, leading to more varied alignments.^[80]

Inequalities

In a right triangle with legs a and b and hypotenuse c, the triangle inequalities a + b > c, a + c > b, and b + c > a hold, but since c > a and c > b, the conditions a + c > b and b + c > a are automatically satisfied, making a + b > c the strictest constraint.$$](https://mathworld.wolfram.com/TriangleInequality.html) The inradius r is given by r = (a + b - c)/2, while the circumradius R = c/2.[

(https://mathworld.wolfram.com/RightTriangle.html) These yield the relation $a + b = c + 2r$, so $c < a + b = c + 2r$, with the first inequality following from the triangle inequality and the second holding with equality.

](https://mathworld.wolfram.com/RightTriangle.html) Moreover, R > r because c/2 > (a + b - c)/2 simplifies to a + b < 2c, which holds since c^2 = a^2 + b^2 \geq ((a + b)/\sqrt{2})^2 by the QM-AM inequality, implying a + b \leq c\sqrt{2} < 2c.[(https://mathworld.wolfram.com/RightTriangle.html)](https://mathworld.wolfram.com/QuadraticMean-ArithmeticMeanInequality.html) The arithmetic mean-geometric mean inequality applied to the legs states that \frac{a + b}{2} \geq \sqrt{ab}, with equality when a = b (the isosceles case).[$$(https://mathworld.wolfram.com/AM-GM Inequality.html) For a fixed perimeter P = a + b + c, the area A = \frac{1}{2}ab of a right triangle is maximized when a = b, corresponding to the isosceles right triangle; however, this maximum is less than that of the equilateral triangle among all triangles with the same perimeter.[](https://people.reed.edu/~zdaugherty/teaching/m201f18/notes/Optimization.pdf)

Coordinate Representation

In analytic geometry, a right triangle is commonly positioned on the Cartesian coordinate plane with its right angle at the origin (0,0), one leg extending along the positive x-axis to the point (a,0), and the other leg along the positive y-axis to the point (0,b), where a > 0 and b > 0 represent the lengths of the legs.^[83] This axis-aligned configuration simplifies calculations involving distances, vectors, and transformations./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/02%3A_Vectors/2.04%3A__Coordinate_Systems_and_Components_of_a_Vector_(Part_1)) The hypotenuse connects the points (a,0) and (0,b), and its length can be verified using the distance formula:

\sqrt{(a - 0)^2 + (0 - b)^2} = \sqrt{a^2 + b^2}.

This computation directly confirms the relationship between the side lengths, aligning with the Pythagorean theorem.^[83] The legs of the triangle can be represented as position vectors \vec{u} = \langle a, 0 \rangle and \vec{v} = \langle 0, b \rangle from the origin. These vectors are orthogonal, as their dot product is \vec{u} \cdot \vec{v} = a \cdot 0 + 0 \cdot b = 0, which characterizes perpendicularity in the plane.^[84] One key application of this representation is computing the area of the triangle, which is \frac{1}{2}ab. Equivalently, the area can be found using the magnitude of the determinant of the matrix formed by the vectors:

\frac{1}{2} \left| \det \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \right| = \frac{1}{2} |ab| = \frac{1}{2}ab,

providing a linear algebra perspective on the geometric area.^[85] This standard placement also facilitates geometric transformations, such as rotations about the origin. For instance, rotating the triangle by an angle \theta counterclockwise applies the rotation matrix to each vertex, transforming (a,0) to (a \cos \theta, a \sin \theta) and (0,b) to (-b \sin \theta, b \cos \theta). Such operations preserve the right angle and enable analysis of the triangle under rigid motions.^[86] While the axis-aligned position is foundational, a right triangle in general position on the plane has its right-angled vertex at arbitrary coordinates (x_0, y_0), with legs extending in non-axial directions, though this requires more complex vector adjustments for similar computations./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/02%3A_Vectors/2.04%3A__Coordinate_Systems_and_Components_of_a_Vector_(Part_1))

References

[1]
Solving Right Triangles & Applications of Static Trigonometry
Quick Intro · A right angle is an angle that measures 90 ∘ . · A right triangle is a triangle with a right angle. · The sum of the angles in a triangle is 180 ∘ .
[2]
[PDF] 6.1 Basic Right Triangle Trigonometry
Hypotenuse: side opposite the right angle, side c in the diagram above. 11. Pythagorean Theorem: 𝑐𝟐 = 𝑎2 + 𝑏2. Example 1: A right triangle has a hypotenuse ...
[3]
The Pythagorean Theorem - Ximera - The Ohio State University
Oct 10, 2025 · When working with a right triangle, we call the two shorter sides of the triangle, or the two sides which are not the hypotenuse, the legs of ...
[4]
MA2C Right Triangles and Trigonometry
Theorem B.1.3. The Pythagorean theorem states that the sum of the squares of the legs of a right triangle must equal the square of the hypotenuse. Consequently ...<|control11|><|separator|>
[5]
Right Triangles - cs.clarku.edu
The Pythagorean theorem will give us the hypotenuse. For instance, if a = 10 and b = 24, then c2 = a2 + b2 = 10 ...
[6]
[PDF] The Pythagorean Theorem - Palm Beach State College
In any right triangle, the sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse: a2 + b2 = ...Missing: properties | Show results with:properties
[7]
[PDF] Right triangles and the Pythagorean Theorem
Remember that the angles of a triangle always add up to 180 degrees. Since a right angle is 90 degrees, that means the other two angles add up to 90 degrees.Missing: definition | Show results with:definition
[8]
Right Triangle Trigonometry - Portland Community College
(The) cosine of theta is equal to the adjacent side's length dividing the hypotenuse's length. (The) tangent of theta is equal to the adjacent side's length ...
[9]
3. Right triangle trigonometry - Pre-Calculus
To check that √3 is right for the base of our triangle, we use the Pythagorean theorem: b2+12=22.
[10]
Appendix L: Triangles and the Pythagorean Theorem – Physics 131
A right triangle is a triangle that has one 90° angle, which is often marked with a ⦜ symbol. Properties of Similar Triangles. If two triangles are similar, ...
[11]
Triangle | Definition & Facts | Britannica
Sep 19, 2025 · A triangle is a three-sided polygon, formed by three line segments, whose end points intersect at points known as the vertices.
[12]
sum of angles of triangle in Euclidean geometry - PlanetMath.org
Sep 26, 2013 · Proof. Let ABC A ⁢ B ⁢ C be an arbitrary triangle with the interior angles α α , β β , γ γ . In the plane of the triangle we set the lines ...Missing: source | Show results with:source
[13]
Right Triangle -- from Wolfram MathWorld
A right triangle is triangle with an angle of 90 degrees (pi/2 radians). The sides a, b, and c of such a triangle satisfy the Pythagorean theorem ...
[14]
Triangle -- from Wolfram MathWorld
A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be ...
[15]
Euclidean geometry | Definition, Axioms, & Postulates - Britannica
Euclidean geometry is the study of plane and solid figures on the basis of axioms and theorems employed by the ancient Greek mathematician Euclid.
[16]
Euclid's Elements, Book I, Proposition 47 - Clark University
The rule for computing the hypotenuse of a right triangle was well known in ancient China. It is used in the Zhou bi suan jing, a work on astronomy and ...
[17]
[PDF] Pythagorean Theorem Proposition 47 of Book
In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
[18]
[PDF] Pythagorean Theorem Using Similar Triangles - Penn Math
Pythagorean Theorem says that. (1) a2 + b2 = c2. This is Euclid's second proof (it uses similar triangles). I am surprised that this classical approach that ...
[19]
Euclid's proof of the Pythagorean Theorem
According to Euclid, the first step of the proof requires us to construct (or "describe") squares on all three sides of the triangle, and in order to do this, ...Missing: statement | Show results with:statement
[20]
Euclid's Proof of the Pythagorean Theorem | Synaptic | Central College
Jan 31, 2019 · Euclid's proof takes a geometric approach rather than algebraic; typically, the Pythagorean theorem is thought of in terms of a² + b² = c², not ...
[21]
Pythagorean Triples
The standard method used for obtaining primitive Pythagorean triples is to use the generating equations, a = r 2 - s 2 , b = 2rs , c = r 2 + s 2 (1) where
[22]
[PDF] Unifications of Pythagorean Triple Schema - Digital Commons@ETSU
Apr 18, 2019 · In order for the triple generated by Euclid's formula to be primitive, both m and n must be coprime and not both odd. Euclid's formula generates.
[23]
[PDF] Proofs of Pythagorean Theorem - OU Math
3 Proof by similar triangles. Let CH be the perpendicular from C to the side ... Using (1) and (2), we rewrite this as c = a2 c. + b2 c. , which is.
[24]
Thales' Theorem -- from Wolfram MathWorld
Thales' Theorem. ThalesTheorem. An inscribed angle in a semicircle is a right angle. See also. Inscribed Angle, Right Angle, Semicircle. Explore with Wolfram| ...
[25]
[PDF] Math 161 - Notes - UCI Mathematics
Ancient Greece (from c. 600 BC) Philosophers such as Thales and Pythagoras began the process of abstraction. General statements (theorems) formulated and proofs ...
[26]
[PDF] Chapter 1. Thales and Pythagoras
Jul 23, 2021 · Herodotus (484 BCE–425 BCE) says that. Thales predicted a solar eclipse of 585 BCE (though some skepticism exists of this claim). Several ...
[27]
Thales of Miletus | Internet Encyclopedia of Philosophy
'Eudemus in his history of geometry attributes the theorem itself to Thales, saying that the method by which he is reported to have determined the distance ...
[28]
[PDF] Circle Geometry - Berkeley Math Circle
Sep 15, 2015 · Proof of Thales' Theorem: Since OA = OB = OC, triangles AOB and AOC are isosceles. The isosceles triangle theorem says that the base angles of ...
[29]
Problem Solver's Toolbox - UW-Math Wiki
Sep 12, 2021 · The converse of Thales's theorem states that if △ A B C is a right triangle with hypotenuse then we can draw a circle with a center that is the ...
[30]
[PDF] thales of Miletus - Princeton University
The converse of Thales's theorem is also true: the locus of all points from which a given line segment sub- tends a constant angle is an arc of a circle ...
[31]
Early Greek - CSUSM
His most impressive result was that any triangle inscribed in a circle whose base is a diameter is a RIGHT TRIANGLE. This is called "Thales Theorem". Thales ...
[32]
Hypotenuse -- from Wolfram MathWorld
The hypotenuse of a right triangle is the triangle's longest side, i.e., the side opposite the right angle. The word derives from the Greek hypo- ("under") ...
[33]
Inequalities in Triangle
6\sqrt{3}r\le a+b+c From the isoperimetric theorem for triangles, \displaystyle \left(\frac{a+b+c}{2}\right)^2 \ge 3\sqrt{3}S=3\sqrt{3}\frac{a+b+c}{2}r, so ...
[34]
Pythagorean Triangles and Triples - Dr Ron Knott
A Pythagorean triple which is not a multiple of another is called a primitive Pythagorean triple. So 3,4,5 and 5,12,13 are primitive Pythagorean triples but ...
[35]
[PDF] triangles in neutral geometry three theorems of measurement lesson ...
note that this means that a triangle cannot support more than one right or obtuse angle– if a triangle has a right angle, or an obtuse angle, then the other ...
[36]
Adjacent Side -- from Wolfram MathWorld
Given an acute angle in a right triangle, the adjacent side is the leg of the triangle from which the angle to the hypotenuse is measured.
[37]
[PDF] Section 4.3, Right Triangle Trigonometry
2 More Trigonometric Properties and Identities. Cofunctions of complementary angles are equal. So, if θ is acute, sin(90◦ − θ) = cos θ cos(90◦ − θ) = sin ...
[38]
A Trigonometric Observation in Right Triangle
ΔABC is right iff sin²A + sin²B + sin²C = 2 · Advanced Identities · Hunting Right Angles · Point on Bisector in Right Angle · Trigonometric Identities with ...
[39]
Area of a triangle - cs.clarku.edu
There are several ways to compute the area of a triangle. For instance, there's the basic formula that the area of a triangle is half the base times the height.Missing: derivation | Show results with:derivation
[40]
[PDF] Pick's Theorem: how to calculate the area of a polygon
May 21, 2025 · ... formula that the area of a right triangle is the base multiplied by the height divided by 2 for each of these right triangles. The right ...<|control11|><|separator|>
[41]
[PDF] Trig 3.2 ~ The Law of Cosines
The area of a triangle in two ways: Area = 1/2 ab sin C or. Area = √s(s-a)(s-b)(s-c) where s = semiperimeter, a+b+c. 2. Find the area of a triangle with sides ...
[42]
Heron's Formula -- from Wolfram MathWorld
An important theorem in plane geometry, also known as Hero's formula. Given the lengths of the sides a, b, and c and the semiperimeter s=1/2(a+b+c) (1) of a ...
[43]
[PDF] Heron's Formula for Triangular Area - Mathematics
In a right-angled triangle, if a perpendicular is drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and.
[44]
[PDF] ORMC AMC Group: Week 6 Geometry: Triangles - UCLA Math Circle
Oct 30, 2022 · Another useful formula is called Heron's Formula: K = ps(s − a)(s ... that a triangle is a right triangle if and only if its sides a ...
[45]
Altitude to the Hypotenuse - CliffsNotes
The altitude drawn to the hypotenuse of a right triangle creates two similar right triangles, each similar to the original right triangle and similar to each ...
[46]
Apollonius's Theorem | Brilliant Math & Science Wiki
Apollonius's theorem is an elementary geometry theorem relating the length of a median of a triangle to the lengths of its sides.
[47]
theory.html
T35 The median to the hypotenuse of any right triangle is half as long as the hypotenuse. Idea of proof . Drop a perpendicular from the midpoint of the ...
[48]
Math 487 Lab 4: Carpenter Locus Theorem
If ABC is a right triangle, with right angle C, prove that the circumcenter of the triangle is the midpoint of the hypotenuse. Or, in other words, prove ...
[49]
The Inradius of a Right Triangle With Integral Sides
The inradius (r) of a right triangle with integral sides is an integer, given by r = y(x - y).
[50]
Inradius of Right Triangle - Expii
Given the above right triangle, the inradius is denoted by a dotted red line. Inradius r can be solved using the following equation: r = 12 (a + b - c).
[51]
Inradius, perimeter, & area (video) - Khan Academy
Aug 11, 2012 · 1/2 times the inradius times the perimeter of the triangle. Or sometimes you'll see it written like this. It's equal to r times P over s-- sorry, P over 2. And ...Missing: implies | Show results with:implies<|control11|><|separator|>
[52]
How do you derive that the inradius in a right triangle is r=a+b−c2?
Oct 19, 2021 · If we have a right triangle then the inradius is equal to r= a+b−c 2 , where c is the hypothenuse and a and b are the legs.Range of inradius of a right Triangle - Math Stack ExchangeIncircle Right Triangle Proof - geometry - Math Stack ExchangeMore results from math.stackexchange.com
[53]
[PDF] 20 concurrence II
1. Thales' Theorem: A triangle 스ABC has a right angle at C if and only if C is on the circle with diameter AB. 2. The diagonals of a parallelogram bisect one ...
[54]
Euler's Inequality - U.C. Berkeley Mathematics
One of the oldest inequalities about triangles is that relating the radii of the circumcircle and incircle. It was proved by Euler and is contained in the ...Missing: derivation | Show results with:derivation
[55]
Excircles -- from Wolfram MathWorld
An excircle is a circle tangent to two extended sides of a triangle and the third side. Every triangle has three excircles.Missing: formula | Show results with:formula
[56]
Exradius -- from Wolfram MathWorld
The radius of an excircle. Let a triangle have exradius r_A (sometimes denoted rho_A), opposite side of length a and angle A, area Delta, and semiperimeter ...
[57]
If an exradius of a triangle is the sum of the two other exradii and the ...
Sep 8, 2016 · In right △ABC, with ∠C=90∘, and standard definitions for a,b,c,s,ra,rb,rc, we have ra=s−brb=s−arc=s. These follow from the fact that the ...
[58]
Euclid's Elements, Book IV, Proposition 5 - Clark University
There is also an equation relating the circumradius R, the inradius r, and the three exradii rA, rB, and rC: 4R = rA + rB + rC – r, and a number of other ...
[59]
Reciprocal Trigonometric Functions | Brilliant Math & Science Wiki
We first explore the reciprocal trigonometric functions by studying the relationships between side lengths in a right triangle.
[60]
Reciprocal trig ratios (article) - Khan Academy
The cosecant is the reciprocal of the sine. It is the ratio of the hypotenuse to the side opposite a given angle in a right triangle. A right triangle ...
[61]
[PDF] 8 3 Practice Special Right Triangles
Practical Uses of 45-45-90 Triangles. These triangles frequently show up in problems involving squares because if you cut a square diagonally, you create two ...
[62]
None
### Summary of 45-45-90 Triangle Properties from https://www.andrews.edu/~adamst/geonotes/Chapt.%2007/7.4.pdf
[63]
45 45 90 Triangle - Housing Innovations
May 21, 2025 · The 45 45 90 triangle is a special right triangle with two 45-degree angles and one 90-degree angle. This triangle is particularly useful in ...
[64]
30-60-90 Triangle -- from Wolfram MathWorld
For a 30-60-90 triangle with hypotenuse of length a, the legs have lengths b = asin(60 degrees)=1/2asqrt(3) (1) c = asin(30 degrees)=1/2a, (2) and the area ...
[65]
Special right triangles review (article) - Khan Academy
The 30-60-90 triangle ratio can also be written relative to the shortest leg as y as the base leg, y√3 as the height, and 2y as the hypotenuse (the key idea of ...
[66]
Trigonometry Angles--Pi/6 -- from Wolfram MathWorld
Construction of the angle pi/6=30 degrees produces a 30-60-90 triangle, which has angles theta=pi/6 and 2theta=pi/3.<|separator|>
[67]
Trigonometry Angles--Pi/3 -- from Wolfram MathWorld
Construction of the angle pi/3=60 degrees produces a 30-60-90 triangle, which has angles theta=pi/3 and theta/2=pi/6.
[68]
30-60-90 Triangles | Properties, Formula & Examples - Study.com
An equilateral triangle, which has three equal sides, can be split into two 30-60-90 triangles by the height from any angle to base. The Pythagorean Theorem ...
[69]
[PDF] The Kepler Triangle and Its Kin - FORMA
The Kepler triangle, also known as the golden right triangle, is the ... He just mentioned it in his letter to Prof Michael Mästlin dated 30th October 1597.
[70]
Kepler Triangle -- from Wolfram MathWorld
The Kepler triangle is triangle with side lengths in proportion phi^(-1/2):1:phi^(1/2) , where phi is the golden ratio. It is therefore a reciprocal proportion ...Missing: definition | Show results with:definition
[71]
Golden Ratio - Definition, Symbol, Formula, & Examples - Math Monks
Nov 13, 2023 · Golden Ratio in Kepler's Triangle. The ratio of these sides is found to be in the 1 : ϕ : ϕ . It inspired Johannes Kepler to create the ...
[72]
Geometry in Art & Architecture Unit 2 - Dartmouth Mathematics
If the pyramid is the Great Pyramid, we get the so-called Egyptian Triangle. It is also called the Triangle of Price, and the Kepler triangle. This triangle is ...
[73]
Phi and Fibonacci in Kepler and Golden Triangles
May 13, 2012 · This orthogonal scalene triangle has all its sides in ratio T and scalene angle ArcTan [ T ] , T=SQRT[Phi]. Its hypotenuse is T^3, its ...
[74]
Euler Line -- from Wolfram MathWorld
The Euler line consists of all points with trilinear coordinates alpha:beta:gamma which satisfy |alpha beta gamma; cosA cosB cosC; cosBcosC cosCcosA cosAcosB|= ...
[75]
Euler line - AoPS Wiki
### Summary of Euler Line in Right-Angled Triangles
[76]
The Euler Line of a Triangle - Clark University
... triangle; but for right triangles, it coincides with the midpoint of the hypotenuse. As Euclid proved in Propsition IV.3 of his Elements, the circumcenter ...
[77]
Right Triangle - Encyclopedia.pub
Oct 17, 2022 · The orthocenter lies on the circumcircle. The distance between the incenter and the orthocenter is equal to 2 r . 3. Trigonometric ...
[78]
Euler Line | Brilliant Math & Science Wiki
The Euler line of a triangle is a line going through several important triangle centers, including the orthocenter, circumcenter, centroid, and center of the ...
[79]
Nine-Point Circle -- from Wolfram MathWorld
The nine-point circle, also called Euler's circle or the Feuerbach circle, is the circle that passes through the perpendicular feet H_A , H_B , and H_C ...
[80]
https://brilliant.org/wiki/euler-line/
[81]
Distance Formula - MathBitsNotebook(Geo)
Start by drawing a right triangle anywhere on a coordinate axes. We will be using the first quadrant for clarity. Draw the Δ so the legs lie on the grids of ...
[82]
2.2 Coordinate Systems and Components of a Vector
In the Cartesian system, the x and y vector components of a vector are the orthogonal projections of this vector onto the x– and y-axes, respectively.
[83]
[PDF] Area and determinants in 2D A B θ - MIT
Figure 2: Using vectors to find the area of a triangle. We start with a triangle in the plane described by vectors A and B. We know that its area is base times ...
[84]
Rotations NOT Centered at the Origin - MathBitsNotebook
The basic answer is that you can use right triangles to rotate the points (Method 1), or you can duplicate the origin to coincide with the new center point.