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Incircle and excircles

In geometry, the incircle of a triangle is the unique circle that lies inside the triangle and is tangent to all three of its sides, with its center known as the incenter, which is the intersection point of the triangle's angle bisectors.^[1] The excircles, in contrast, are three distinct circles, each lying outside the triangle and tangent to one side and to the extensions of the other two sides, with their centers called excenters, formed by the intersection of one internal angle bisector and two external angle bisectors.^[2] These circles are fundamental in triangle geometry, providing insights into tangency points, radii calculations, and related concurrency points.^[3] The radius of the incircle, termed the inradius ([r](/page/R)), is given by the formula [r](/page/R) = A / s, where A is the area of the triangle and s is its semiperimeter (s = (a + b + c)/2, with a, b, c as side lengths).^[4] For the excircles, the radii (exradii) are r_a = A / (s - a), r_b = A / (s - b), and r_c = A / (s - c), each corresponding to the excircle opposite vertex A, B, or C, respectively.^[2] The area of the triangle can also be expressed as A = r \cdot s, highlighting the incircle's role in area computations, while the sum of the exradii satisfies r_a + r_b + r_c - r = 4R, where R is the circumradius.^[4] Notable properties include the tangency points of the incircle forming the contact triangle, whose cevians concur at the Gergonne point, and the excircle tangency points concurring at the Nagel point.^[1] The incircle is tangent to the nine-point circle of the triangle.^[1] These elements extend to applications in tangential polygons and advanced triangle centers, underscoring their importance in Euclidean geometry.^[2]

Incircle and Incenter

Definition of Incircle and Incenter

The incircle of a triangle is the unique circle that lies entirely within the triangle and is tangent to all three sides. It is also known as the inscribed circle and represents the largest circle that can fit inside the triangle while touching each side at exactly one point. This circle is a fundamental element in triangle geometry, providing insights into the triangle's internal structure and properties. The center of the incircle, termed the incenter, is the point of concurrency of the triangle's three angle bisectors. Each angle bisector divides the corresponding angle into two equal parts, and their intersection forms the incenter, which is equidistant from all three sides; this common distance is the inradius. The incenter thus serves as the geometric center of the incircle and plays a key role in various constructions and theorems related to triangle symmetry. The concepts of the incircle and incenter were first explored by ancient Greek mathematicians, with Euclid formalizing their properties in his Elements around 300 BCE. These early studies emphasized the incircle's role in balancing tangential contacts and area computations. For a triangle with side lengths a, b, and c opposite vertices A, B, and C respectively, the semiperimeter is defined as s = \frac{a + b + c}{2}. The points where the incircle touches the sides divide each side into two segments: the lengths of the tangents from vertex A to the points of tangency on sides AB and AC are both s - a; similarly, from B they are s - b, and from C they are s - c. This equal-tangent property ensures the incircle's balanced positioning and aids in visualizing its placement relative to the triangle's vertices.

Coordinate Representations of Incenter

The incenter of a triangle can be precisely located using various coordinate systems, each offering distinct advantages for geometric computations and proofs. These systems include trilinear, barycentric, and Cartesian coordinates, which facilitate the analysis of the incenter's position relative to the triangle's vertices and sides. In trilinear coordinates, the incenter is represented as (1:1:1).^[5] These coordinates are homogeneous, meaning they are defined up to scalar multiplication, and they correspond to the signed distances from the point to the triangle's sides, normalized relative to the side lengths. For the incenter, the equal distances to all three sides (equal to the inradius) result in this symmetric form, making trilinear coordinates particularly suited for problems involving perpendicular distances and cevian intersections.^[6] Barycentric coordinates provide another homogeneous representation of the incenter as (a:b:c), where a, b, and c denote the lengths of the sides opposite vertices A, B, and C, respectively.^[5] This form arises from the relation between trilinear and barycentric systems: the barycentric coordinates are obtained by multiplying the trilinear coordinates by the corresponding side lengths, yielding (a·1 : b·1 : c·1) = (a:b:c).^[7] The derivation stems from viewing the incenter as the center of mass of the triangle's vertices weighted by the opposite side lengths, reflecting the balance achieved at the intersection of the angle bisectors.^[8] Specifically, if A, B, and C are the position vectors of the vertices, the incenter I satisfies I = (aA + bB + cC) / (a + b + c).^[5] In Cartesian coordinates, assuming the triangle has vertices A(x_A, y_A), B(x_B, y_B), and C(x_C, y_C), the incenter's position is given by

I_x = \frac{a x_A + b x_B + c x_C}{a + b + c}, \quad I_y = \frac{a y_A + b y_B + c y_C}{a + b + c}.

This formula directly extends the barycentric representation to Euclidean space, allowing for numerical computation and visualization in a plane.^[5] Each coordinate system has specific computational benefits: trilinear coordinates excel in derivations involving side distances and homogeneous properties, barycentric coordinates are advantageous for mass point geometry and affine-invariant calculations (such as balancing cevians in the incenter example with weights proportional to side lengths), while Cartesian coordinates are ideal for direct metric computations and plotting in standard geometry software.^[9]

Inradius Formula and Derivation

The inradius r of a triangle is the radius of its incircle, which touches all three sides internally. It is given by the formula r = \frac{A}{s}, where A is the area of the triangle and s = \frac{a + b + c}{2} is the semiperimeter, with a, b, and c denoting the side lengths.^[4]^[10] To derive this formula, consider the incenter I, the center of the incircle. The points of tangency divide the sides into segments equal to the tangent lengths from each vertex. The triangle's area A can be decomposed into the three smaller triangles formed by connecting I to the vertices: \triangle AIB, \triangle BIC, and \triangle CIA. However, a more straightforward approach uses the tangential regions: the area is the sum of the areas of three right triangles (or sectors in a limiting sense, but precisely via perpendiculars) from I to each side. Each such region has height r (the perpendicular distance from I to the side) and base equal to the side length, yielding

A = \frac{1}{2} r a + \frac{1}{2} r b + \frac{1}{2} r c = r \left( \frac{a + b + c}{2} \right) = r s.

Solving for r gives r = \frac{A}{s}. This derivation relies on the property that the incircle is tangent to all sides, ensuring equal perpendicular distances.^[10]^[11]^[12] An alternative expression for the inradius is r = (s - a) \tan \frac{A}{2}, where A is the angle at vertex A opposite side a. To derive this, note that the lengths of the tangents from vertex A to the points of tangency on sides AB and AC are both s - a. The angle bisector from A passes through I, splitting \angle A into two equal angles of \frac{A}{2}. Consider the right triangle formed by vertex A, the point of tangency on AB, and the foot of the perpendicular from I to AB: the adjacent side to \frac{A}{2} is s - a, and the opposite side is r, so \tan \frac{A}{2} = \frac{r}{s - a}, hence r = (s - a) \tan \frac{A}{2}. Similar expressions hold for the other angles.^[4]^[13] This relation A = r s integrates historically with Heron's formula for the area, A = \sqrt{s(s - a)(s - b)(s - c)}, attributed to Heron of Alexandria in the 1st century CE. Substituting yields r = \frac{\sqrt{s(s - a)(s - b)(s - c)}}{s} = \sqrt{\frac{(s - a)(s - b)(s - c)}{s}}, providing a side-length-only expression for r without explicit area computation. Early proofs of Heron's formula, such as those using cyclic quadrilaterals or trigonometric identities, often leverage the inradius to bridge perimeter and area concepts, emphasizing the incircle's role in partitioning the triangle's area into equal-tangent components.^[4]^[14] For an equilateral triangle with side length a, the area is A = \frac{\sqrt{3}}{4} a^2 and s = \frac{3a}{2}, so r = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2}} = \frac{\sqrt{3}}{6} a. This illustrates the formula's application, where the inradius is one-third of the height \frac{\sqrt{3}}{2} a.^[4]

Properties of Incircle Touch Points

The points where the incircle of a triangle touches the sides are known as the points of tangency. For a triangle ABC with side lengths a = BC, b = AC, c = [AB](/page/AB), and semiperimeter s = (a + b + c)/2, denote the points of tangency on sides BC, CA, and AB by X, Y, and Z, respectively. The lengths of the tangent segments from each vertex to the points of tangency are equal, a property arising from the fact that tangents drawn from a common external point to a circle are congruent.^[15] Thus, the distance from vertex A to Z (on AB) equals the distance from A to Y (on AC), denoted x = s - a. Similarly, the distance from B to Z (on AB) equals the distance from B to X (on BC), denoted y = s - b, and the distance from C to X (on BC) equals the distance from C to Y (on AC), denoted z = s - c. These relations follow directly from solving the system of equations for the side lengths: c = x + y, b = x + z, a = y + z, yielding x = (b + c - a)/2 = s - a, and analogously for y and z.^[15] The positions of the touch points on the sides can thus be specified: X divides BC such that BX = y = s - b and CX = z = s - c; Y divides CA such that CY = z = s - c and AY = x = s - a; Z divides AB such that AZ = x = s - a and BZ = y = s - b. These distances represent the segments from the vertices to the nearest touch points along the adjacent sides.^[15] The three touch points X, Y, and Z form a triangle known as the contact triangle (or intouch triangle), which is perspective to the reference triangle ABC. The sides of the contact triangle are tangent to the incircle at X, Y, and Z, and its vertices lie on the sides of ABC.^[16] The cevians joining each vertex of ABC to the opposite touch point (AX, BY, CZ) are concurrent at the Gergonne point.

Excircles and Excenters

Definition of Excircles and Excenters

In a triangle, an excircle (or escribed circle) is a circle that lies outside the triangle and is tangent to one of its sides and to the extensions of the other two sides. Every triangle has three distinct excircles, one opposite each vertex; for instance, the excircle opposite vertex A (denoted the A-excircle) is tangent to side BC at a point and to the infinite extensions of sides AB and AC beyond points B and C, respectively. The excenter of an excircle is its center, which serves as the point of concurrency for specific angle bisectors of the triangle. The three excenters, labeled I_a, I_b, and I_c opposite vertices A, B, and C respectively, are each formed by the intersection of the internal angle bisector from one vertex and the external angle bisectors from the other two. Specifically, I_a lies at the intersection of the internal bisector of \angle A and the external bisectors of \angle B and \angle C.^[17] Unlike the incircle, which touches all three sides internally from within the triangle, each excircle touches one side directly and the extensions of the others from the exterior, resulting in a larger radius and positioning the circle outside the triangle opposite the associated vertex. This external configuration distinguishes the excircles as counterparts to the internal incircle in the study of tangential circles to a triangle.

Coordinate Representations of Excenters

The excenters of a triangle can be expressed using trilinear coordinates, which are homogeneous coordinates proportional to the directed distances from the point to the sides of the triangle. The excenter opposite vertex A, denoted I_a, has trilinear coordinates (-a : b : c), where a, b, and c are the lengths of the sides opposite vertices A, B, and C respectively. The negative sign for the a-coordinate reflects the external position of I_a relative to side BC, as it lies outside the triangle on the extension of the angle bisector from A. Similarly, the excenter I_b opposite B has coordinates (a : -b : c), and I_c opposite C has (a : b : -c).^[18] Barycentric coordinates provide another homogeneous representation for the excenters, closely related to trilinear coordinates by normalization with respect to the triangle's area. For I_a, the barycentric coordinates are (-a : b : c), interpreted as signed masses placed at the vertices: a negative mass at A and positive masses at B and C, whose center of mass yields the excenter. To obtain normalized barycentric coordinates summing to 1, divide by the total weight: the coordinates become \left( \frac{-a}{-a+b+c}, \frac{b}{-a+b+c}, \frac{c}{-a+b+c} \right). The cyclic permutations apply analogously for I_b and I_c, with the negative sign indicating the vertex opposite the excircle's internal tangency.^[19] In Cartesian coordinates, the position of an excenter follows directly from the barycentric representation as a weighted average of the vertices' positions. For I_a, with vertices A = (A_x, A_y), B = (B_x, B_y), and C = (C_x, C_y), the coordinates are

I_a = \left( \frac{-a A_x + b B_x + c C_x}{-a + b + c}, \frac{-a A_y + b B_y + c C_y}{-a + b + c} \right).

This formula arises by applying the signed weights from the barycentric coordinates to the vertex positions and normalizing by the sum of the weights, providing an explicit embedding in the plane. The same weighted average structure holds for I_b and I_c with their respective sign flips.^[19] The three excenters I_a, I_b, and I_c form the vertices of the excentral triangle, a triangle whose orthocenter is the incenter of the original triangle.^[20]

Exradii Formulas and Derivation

The exradii of a triangle are the radii of its excircles, denoted r_a, r_b, and r_c, opposite vertices A, B, and C respectively. Let \Delta be the area of the triangle and s its semiperimeter. The formulas are r_a = \frac{\Delta}{s - a}, r_b = \frac{\Delta}{s - b}, and r_c = \frac{\Delta}{s - c}, where a, b, and c are the side lengths opposite A, B, and C.^[21] To derive these, consider the excircle opposite A, tangent to side BC internally and to the extensions of AB and AC externally. The excenter I_a forms three tangential triangles with the sides: I_aBC (internal tangent), and I_aAB, I_aAC (external tangents). Consider the areas of the tangential triangles. The area \Delta of \triangle ABC equals the area of \triangle I_aAB plus the area of \triangle I_aAC minus the area of \triangle I_aBC. Each area is \frac{1}{2} times the base times the height r_a, yielding \Delta = \frac{1}{2} r_a (b + c - a) = r_a (s - a). Thus, r_a = \frac{\Delta}{s - a}; the other exradii follow cyclically.^[22]^[21] An alternative expression is r_a = s \tan\frac{A}{2}, with cyclic permutations for r_b and r_c. To derive this using the extended law of sines, note that the excenter I_a lies on the internal angle bisector of \angle A. Consider the right triangle formed by I_a, the touch point on the extension of AB, and the foot of the perpendicular from I_a to AB. The adjacent side to \angle A/2 (half-angle along the bisector) is the tangent length from A to the touch point, which equals s, and the opposite side is r_a. Thus, \tan\frac{A}{2} = \frac{r_a}{s}, so r_a = s \tan\frac{A}{2}. The extended law of sines confirms consistency via a = 2R \sin A, linking to half-angle identities, but the bisector geometry provides the direct proof.^[21] The exradii relate to the inradius r by \frac{1}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. This follows from the area expressions: \frac{1}{r_a} = \frac{s - a}{\Delta}, and summing yields \frac{(s - a) + (s - b) + (s - c)}{\Delta} = \frac{3s - (a + b + c)}{\Delta} = \frac{s}{\Delta} = \frac{1}{r}.^[21] For example, in a right triangle with sides 3, 4, 5 (s = 6, \Delta = 6, r = 1), assume the right angle at C opposite the hypotenuse c = 5, with a = 4, b = 3; the exradii are r_a = 3, r_b = 2, r_c = 6. These exceed r, with r_c largest opposite the longest side, and satisfy \frac{1}{1} = \frac{1}{3} + \frac{1}{2} + \frac{1}{6}. For the right angle C = 90^\circ, r_c = 6 \tan(45^\circ) = 6.^[21]

Properties of Excircle Touch Points

The A-excircle of triangle ABC, with sides a = BC, b = AC, c = AB, and semiperimeter s = (a + b + c)/2, touches side BC internally at point X_a such that the distance from B to X_a is s - c and from C to X_a is s - b.^[23] This placement ensures that the tangent segments from B and C to the point of tangency are equal, consistent with the properties of tangents from a point to a circle. The A-excircle touches the extension of side AB beyond B at point Z_a and the extension of side AC beyond C at point Y_a. The distance from B to Z_a is s - c, so the distance from A to Z_a along the extension is c + (s - c) = s; similarly, the distance from C to Y_a is s - b, so the distance from A to Y_a is b + (s - b) = s.^[23] These tangent lengths from vertex A equal the semiperimeter s, while the lengths from B and C are s - c and s - b, respectively; on extensions, distances are measured positively outward from the vertices, with signed lengths accounting for direction in coordinate geometries (positive for extensions beyond B and C, negative if considering directions toward A). The touch points of the three excircles with the sides of the triangle—one per side—form the extouch triangle, also known as the extangents triangle, which exhibits properties such as being the cevian triangle of the Nagel point and having side lengths derived from the original triangle's dimensions, like a' = \sqrt{a^2 - bc \sin^2 A} for the side opposite the vertex corresponding to side a.^[24] This triangle arises directly from the excircle tangency points on the sides and encapsulates semiperimeter relations, where the positions divide the perimeter such that the arc lengths along the sides relate to s - a, s - b, and s - c in aggregate. These excircle touch points play a key geometric role in constructing tangential quadrilaterals, as the positions on the side extensions and the internal touch allow the excircle to serve as the incircle for a quadrilateral formed by the three triangle sides and an additional tangent line connecting the external touch points, enabling derivations of quadrilateral properties via triangle excircle tangencies.^[23] The semiperimeter s governs these lengths, with the total effective tangent path from A across both extensions equaling $2s, underscoring the excircle's relation to the triangle's perimeter.

Gergonne and Nagel Points and Triangles

The Gergonne point of a triangle is the point of concurrency of the three cevians joining each vertex to the point of tangency of the incircle with the opposite side.^[25] This concurrency follows from Ceva's theorem applied to the side divisions induced by the touch points, where the ratios satisfy (s-b)/ (s-c) · (s-c)/(s-a) · (s-a)/(s-b) = 1, with s denoting the semiperimeter.^[25] In barycentric coordinates with respect to the reference triangle, the Gergonne point has coordinates ((s-b)(s-c) : (s-c)(s-a) : (s-a)(s-b)), or equivalently (1/(s-a) : 1/(s-b) : 1/(s-c)) up to scalar multiple.^[25] The Gergonne triangle, also known as the intouch triangle or contact triangle, is the triangle formed by connecting the three points of tangency of the incircle with the sides of the reference triangle.^[26] This triangle is perspective to the reference triangle, with the Gergonne point serving as the perspector; the lines joining corresponding vertices pass through this concurrency point.^[25] The Nagel point is defined analogously as the point of concurrency of the cevians from each vertex to the point of tangency of the excircle opposite that vertex with the opposite side (extended if necessary).^[27] Ceva's theorem again confirms this concurrency, with the relevant ratios (s-a)/s · s/(s-b) · (s-b)/(s-a) = 1, though adjusted for the external divisions.^[25] Its barycentric coordinates are (s-a : s-b : s-c).^[25] The Nagel point is also known as the isotomic conjugate of the incenter and relates to the splitters, which are the cevians to the excircle touch points.^[27] The Nagel triangle, or extouch triangle, is formed by the three points where the excircles touch the sides of the reference triangle (on extensions for the external tangencies).^[28] Like the Gergonne triangle, it is perspective to the reference triangle, with the Nagel point as the perspector.^[25] The Gergonne and Nagel points are isotomic conjugates of each other, as their barycentric coordinates are reciprocals: the transformation (x : y : z) \mapsto (1/x : 1/y : 1/z) maps one to the other.^[25] In an equilateral triangle, both points coincide with the centroid, incenter, orthocenter, and other classical centers, all lying on the Euler line.^[25] The distance between the Gergonne and Nagel points can be expressed using barycentric distance formulas, yielding a quantity dependent on the side lengths a, b, c and semiperimeter s, though it establishes no unique geometric invariant beyond their conjugate relation.^[29]

Incentral and Excentral Triangles

The incentral triangle is formed by the incenter I and the three excenters I_a, I_b, I_c of a reference triangle ABC. These four points constitute an orthocentric system, in which each point serves as the orthocenter of the triangle formed by the other three.^[30] In this system, the reference triangle ABC functions as the orthic triangle of the excentral triangle (the triangle formed by the three excenters), with the altitudes from the excenters to the opposite sides landing at the vertices of ABC.^[20] The excentral triangle, denoted ΔI_aI_bI_c, has vertices at the three excenters of ABC. It is always acute-angled, with angles measuring 90° - A/2 at I_a, 90° - B/2 at I_b, and 90° - C/2 at I_c.^[31] The orthocenter of the excentral triangle coincides with the incenter I of the original triangle ABC.^[32] Its side lengths are given by I_bI_c = 4R \cos(A/2), I_aI_c = 4R \cos(B/2), and I_aI_b = 4R \cos(C/2), where R is the circumradius of ABC; these can be related to the exradii r_a, r_b, r_c via the formula r_a = 4R \sin(A/2) \cos(B/2) \cos(C/2), though direct expressions in terms of exradii alone are more complex.^[32] The excentral triangle's incircle has radius 2R (\sin(A/2) + \sin(B/2) + \sin(C/2) - 1), where R is the circumradius of ABC, establishing its scale relative to the original triangle.^[33] The excentral triangle is homothetic to the intouch triangle (the contact triangle of the incircle of ABC), with the center of homothety being the isogonal conjugate of the Mittenpunkt; this transformation highlights similarities in their cevian structures and tangency properties.^[34]

Feuerbach Point and Nine-Point Circle Relations

The Feuerbach point of a triangle is the point of tangency between the incircle and the nine-point circle, where the incircle touches the nine-point circle internally.^[35] This point was identified as part of Feuerbach's theorem, published by Karl Wilhelm Feuerbach in his 1822 work Eigenschaften des Dreiecks, which establishes the tangential relations among the incircle, excircles, and nine-point circle.^[36] Feuerbach's theorem states that the nine-point circle is internally tangent to the incircle at the Feuerbach point and externally tangent to each of the three excircles at distinct points, known collectively as forming the Feuerbach triangle.^[35] These tangency points exhibit specific geometric properties, with the Feuerbach point serving as a triangle center, denoted X(11) in the Encyclopedia of Triangle Centers.^[35] In barycentric coordinates with respect to the reference triangle, the Feuerbach point has coordinates a(1 - \cos(B - C)) : b(1 - \cos(C - A)) : c(1 - \cos(A - B)), where a, b, c are the side lengths opposite angles A, B, C respectively.^[35] An alternative algebraic form is (b + c - a)(b - c)^2 : (c + a - b)(c - a)^2 : (a + b - c)(a - b)^2.^[35] The points of tangency with the excircles, while separate, share analogous properties and lie on the Feuerbach triangle, which connects these external tangencies.^[35] Feuerbach's theorem provides historical context for these tangencies, originating from Feuerbach's systematic study of triangle properties in 1822, including proofs of the nine-point circle's contacts with the incircle and excircles using properties of the orthocenter and midpoints.^[36] This theorem highlights concurrencies in triangle geometry, such as the alignment of the nine-point center with other key points.^[37] These relations extend to broader configurations, including the Euler line, where the nine-point center—midpoint of the segment joining the orthocenter and circumcenter—facilitates the tangential properties of the Feuerbach point and related tangencies.^[38] In circle packings, the incircle, excircles, and nine-point circle form a tangential system that influences other packs, such as those involving the intouch triangle, where the Feuerbach point aligns with the Euler line of the intouch triangle itself.^[39]

Advanced Formulas and Theorems

Equations of the Four Circles

The incircle and excircles of a triangle can be described by their equations in Cartesian coordinates using the positions of their centers and radii. The general equation of a circle in the plane is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. For a triangle with vertices A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), and side lengths a = BC, b = [AC](/page/AC), c = AB opposite these vertices respectively, the incenter I has Cartesian coordinates derived from its barycentric coordinates (a : b : c):

I_x = \frac{a x_1 + b x_2 + c x_3}{a + b + c}, \quad I_y = \frac{a y_1 + b y_2 + c y_3}{a + b + c}.

The equation of the incircle is then (x - I_x)^2 + (y - I_y)^2 = r^2, where r is the inradius.^[35] The excenters similarly arise from signed barycentric coordinates. The excenter I_a opposite vertex A has barycentric coordinates (-a : b : c), yielding Cartesian coordinates

(I_a)_x = \frac{-a x_1 + b x_2 + c x_3}{-a + b + c}, \quad (I_a)_y = \frac{-a y_1 + b y_2 + c y_3}{-a + b + c},

and its equation is (x - (I_a)_x)^2 + (y - (I_a)_y)^2 = r_a^2, where r_a is the exradius opposite A. Analogous forms hold for the excenters I_b (barycentrics (a : -b : c)) and I_c (barycentrics (a : b : -c)), incorporating the negative weight for the opposite side into the denominator and numerator of the weighted averages.^[35]^[19] These four circles admit a unified parametric representation through their centers' barycentric coordinates (\epsilon_a a : \epsilon_b b : \epsilon_c c), where each \epsilon_i = \pm 1: all positive for the incenter, and negative for the coordinate corresponding to the opposite vertex in each excenter. The Cartesian coordinates of any such center are obtained by normalizing these barycentrics as weighted averages of the vertex positions, with the circle equation following from the appropriate radius. This signed weighting reflects the internal tangency of the incircle versus the external tangency of the excircles to one side.^[19]

Euler's Distance Formula

Euler's distance formula gives the squared distance between the circumcenter O and the incenter I of a triangle as d^2(O, I) = R(R - 2r), where R is the circumradius and r is the inradius.^[40] This relation highlights the geometric interplay between the triangle's circumcircle and incircle centers. The formula extends to the excenters, with the squared distance between the circumcenter O and the excenter I_a (opposite vertex A) given by d^2(O, I_a) = R(R + 2r_a), where r_a is the exradius opposite A.^[41] Similar expressions hold for the other excenters I_b and I_c. Related distances include that between the incenter I and an excenter I_a, which satisfies d(I, I_a) = 4R \sin\frac{A}{2}.^[42] This form arises from coordinate geometry or trigonometric identities linking the radii and angles. Historically, the formula is attributed to Leonhard Euler's work on triangle centers in 1765, though an earlier discovery by Robert Chapple in 1746 is noted; modern proofs often employ complex number representations of triangle points for elegance and brevity.^[43]

Derivation

One derivation of d^2(O, I) = R(R - 2r) uses trigonometric identities. The distance can be expressed as OI^2 = R^2 (1 - 8 \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2}). Since the inradius r = 4R \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2}, substituting yields OI^2 = R^2 - 2 R r.^[40] For the excenter extension, a similar trigonometric approach applies, replacing the inradius with the exradius in the identity, leading to the positive sign due to the external angle bisectors. Vector formulations position I = \frac{a\mathbf{A} + b\mathbf{B} + c\mathbf{C}}{a+b+c} and compute the norm relative to O, confirming the result after algebraic simplification.^[44] Although the nine-point circle relates the circumcenter to the orthocenter, its midpoint (the nine-point center) aids indirect derivations via Euler line properties, but the direct trigonometric or vector methods are more straightforward for the incenter-excenter distances.^[45]

Generalizations to Polygons and Topological Figures

A tangential polygon, also known as an inscriptible or circumscriptible polygon, is one that possesses an incircle tangent to all its sides. Such polygons exist when the sums of the lengths of every other side are equal, generalizing the condition for triangles where the incircle is always present. The inradius r of a tangential polygon with area A and semiperimeter s is given by r = A / s.^[4] For regular n-gons, the incircle's radius coincides with the apothem, the distance from the center to a side, providing a straightforward geometric interpretation. However, excircles—circles tangent to one side and the extensions of the remaining sides—are primarily defined for triangles and do not generalize directly to polygons with more than three sides in the same unique manner. Analogs known as escribed circles can be constructed for certain polygons, tangent to one side and the extensions of the others, but their existence depends on specific side length conditions and is not guaranteed for all tangential polygons. For instance, convex pentagons and higher can be designed to admit at least one such circle, though this requires tailored constructions rather than a universal property.^[3] In higher-dimensional Euclidean spaces, the concepts extend to simplices, where an insphere is tangent to all facets of an n-simplex. Every simplex admits a unique insphere, with its center (incenter) being the point equidistant from all facets, located at the intersection of the angle bisectors in the appropriate sense or via barycentric coordinates weighted by facet areas. Exspheres analogously exist, each tangent to one facet and the extensions of the others, up to n+1 such spheres for an n-simplex. These structures are utilized in computational geometry, particularly in Delaunay triangulations, where the incircle test—verifying if a circle through three points contains no other points—determines triangulation edges, linking incircles to Voronoi diagrams as dual constructs.^[46]^[47]^[48] Beyond Euclidean geometry, incircles and excircles generalize to non-Euclidean settings, such as hyperbolic triangles, which always possess an incircle tangent to all three sides, with the incenter at the intersection of the angle bisectors. Up to three excircles may also exist, depending on the triangle's configuration, determined similarly by internal and external bisectors. In metric spaces or topological figures, such as topologically embedded simplices, inscribed spheres can be defined via tangency conditions adapted to the ambient geometry, though uniqueness may fail in curved or discrete spaces without additional constraints. These generalizations appear in applications like hyperbolic tilings and computational simulations of curved manifolds.^[49]

References

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Incircle -- from Wolfram MathWorld
An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each of the polygon's sides. The center I of the incircle is called the ...
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Incircles and Excircles | Brilliant Math & Science Wiki
X · X, Y · and ; Z · Z · be the perpendiculars from the incenter to each of the sides. The incircle is the inscribed circle of the triangle that touches all three ...
[3]
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.
[4]
Inradius -- from Wolfram MathWorld
Inradius ; r=1/2acot(pi/n). (1) ; k=(2Delta)/(a+b+c)=Delta/. (5) ; r=k=Delta/s,. (6) ; R^2-d^2=2Rr,. (10) ; 1/(R-d)+1/(R+d)=1/. (11) ...
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Excenter -- from Wolfram MathWorld
The center J_i of an excircle. There are three excenters for a given triangle, denoted J_1, J_2, J_3. The incenter I and excenters J_i of a triangle are an ...
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Incenter -- from Wolfram MathWorld
... triangle are equal. It has trilinear coordinates 1:1:1, i.e., triangle center function alpha_1=1, (1) and homogeneous barycentric coordinates (a,b,c). It is...
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Trilinear Coordinates -- from Wolfram MathWorld
Trilinear coordinates are denoted alpha:beta:gamma or (alpha,beta,gamma) and also are known as homogeneous coordinates or "trilinears."
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Incenter of Triangle in 3D - geometry - Math Stack Exchange
Apr 4, 2014 · The incenter has trilinear coordinates 1:1:1, which you can convert to barycentric coordinates a:b:c, where a=‖B−C‖,b=‖C−A‖,c=‖A−B‖ are the ...
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[PDF] The Conformal Center of a Triangle or Quadrilateral
Now the incenter is the center with barycentric coordinates proportional to the lengths of the opposite sides, which we call the “side-weighted center.” Another ...
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[PDF] Mass Point Geometry (Barycentric Coordinates) - Berkeley Math Circle
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Derivation of Formula for Radius of Incircle - MATHalino
The radius of incircle is given by the formula $r = \dfrac{A_t}{s}$ where At = area of the triangle and s = semi-perimeter.
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https://www.khanacademy.org/math/geometry-home/triangle-properties/angle-bisectors/v/inradius-perimeter-and-area
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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 ...
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Important Solutions Of Triangle Formulas For JEE - BYJU'S
Radius of the incircle: (i) r = Δ/s. (ii) r = (s-a) tan (A/2) = (s-b) tan(B/2) ... 2 + c 2 ). Up Next: Important Integration Formulas For JEE Maths. Test ...
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[PDF] Proof of Heron's Formula - Jamie York Academy
Theorem B: Area of a triangle = r · s; where r is the radius of the inscribed circle, and s is the semi-perimeter of the triangle. Proof: Draw angle bisectors ...
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Property of Points Where In- and Excircles Touch a Triangle
By simple algebra, the distance from a triangle vertex to the tangency point with the incircle equals to the difference between the semiperimeter and the ...
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Incircles and Excircles in a Triangle
Incircle tangency points define cevians meeting at Gergonne point. Excircle tangency points meet at Nagel's point, and these points are isotomic conjugates.
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[PDF] Notes on trilinear coordinates of a triangle
In[18]:= incenter = -Meet[bisector[a], bisector[b]]. Out[18]= 81, 1, 1<. The inscribed circle meets the triangle sides on radii parallel to the altitudes. The ...
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[PDF] Barycentric Coordinates for the Impatient - Evan Chen
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The lines joining the excenters are none other than the bisectors of the external angles and thus pass through the vertices of the given triangle. Let the ...
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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 ...
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From Heron's formula to Descartes' circle theorem - Euler, Erdős
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If the vertices of a triangle are ABC and the excircle touching side BC does so at point D, then AB+BD = AC+CD. (This is easily proved using the theorem ...
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Extouch Triangle -- from Wolfram MathWorld
The extouch triangle DeltaT_1T_2T_3 is the triangle formed by the points of tangency of a triangle DeltaA_1A_2A_3 with its excircles J_1, J_2, and J_3.Missing: properties | Show results with:properties
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[PDF] Barycentric Coordinates: Formula Sheet
- Gergonne Point = X(7) = Ge = [(c + a - b)(a + b - c)]. - Nagel Point = X(8) = Na = [(b + c - a)]. - Mittenpunky = X(9) = [a(b + c - a)]. - Spieker Center ...
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[PDF] Nagel point of the triangle
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[PDF] ON THE GERGONNE AND NAGEL POINTS FOR A HYPERBOLIC ...
The point. Ge, defined by (15), is called the Gergonne point of the triangle ABC. It. Page 4. On the Gergonne and Nagel points for a hyperbolic triangle. 115 is ...
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Barycentric Coordinates
Divide all coordinates by sa sb sc to get this form. ( sb sc : sa sc : sa sb ) = ( 1/sa : 1/sb : 1/sc ) = ( : 1/sb : ). Gergonne point coordinates. Incenter ...
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The incenter is the identity of the group of triangle centers under trilinear multiplication defined by (x : y : z)*(u : v : w) = xu : yv : zw. A ...
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Trigonometry/Circles and Triangles/The Excentral Triangle - Wikibooks
The angles of this triangle are 90 0 − A 2 , 90 0 − B 2 and 90 0 − C 2 {\displaystyle 90^{0}-{\frac {A}{2}},90^{0}-{\frac {B}{2}}{\text{ and }}90^{0}-{\frac {C}{ ...
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Excentral Triangle -- from Wolfram MathWorld
The excentral triangle, also called the tritangent triangle, of a triangle DeltaABC is the triangle J=DeltaJ_AJ_BJ_C with vertices corresponding to the ...
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May 6, 2016 · The excentral triangle's angles are π−A2, π−B2, π−C2; sides are 4RcosA2, 4RcosB2, 4RcosC2; circumradius is 2R; inradius is 2R(∑cycsinA2−1); ...
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ENCYCLOPEDIA OF TRIANGLE CENTERS
Below is a merged response that consolidates all the information from the provided segments into a single, comprehensive summary. To maximize detail and clarity, I will use tables in CSV format where appropriate to present the barycentric and trilinear coordinates, along with additional details like sources and useful URLs. The response will retain all key information while avoiding redundancy and ensuring a dense, organized presentation.
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Distance between incenters and excenters - Math Stack Exchange
Aug 2, 2015 · Let I1,I2,I3 be the excenters opposite to A,B,C. As BI2,CI3 intersect at I. So ∠I2II3=∠BIC=π−B2−C2,because they are vertically opposite angles.Finding the distance between incenter and centroid of a triangleDistance between triangle incenter and verticesMore results from math.stackexchange.com
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In 1746, Chapple [1] discovered the formula d2=R(R−2r), where R, r, d are the circumradius, inradius, and distance between the circumcenter and the incenter ...Missing: original | Show results with:original
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[PDF] Introduction to the Geometry of the Triangle - M∀TH Workout
... Gergonne point appears in ETC as the point X7. 10The Nagel point appears in ETC as the point X8. Page 36. Chapter 3: Homogeneous Barycentric Coordinates. 31.
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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 dropped ...Missing: 1822 paper
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(Not recommended) Incenters of specified simplices - MATLAB
IC = incenters(TR) returns the coordinates of the incenter for each simplex in the triangulation. The incenter associated with simplex i is the i 'th row of IC.
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(PDF) Inscribing Spheres in Topologically Embedded Simplices
Aug 20, 2024 · Abstract. In plane geometry, it is well-known that one can inscribe a circle in any triangle. ... of the n-simplex. The uniqueness of inscribed ...
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[PDF] CMSC 754 Computational Geometry
Can be generalized to computing Voronoi diagrams of line segments fairly easily. Reduction to convex hulls: (For DT.) Computing a Delaunay triangulation of ...
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incircles and excircles - HYPERBOLIC GEOMETRY
(3) tanh(R) = Φ/4cos(½α)cos(½β)cos(½γ). proof. exradius theorem. If the hyperbolic ΔABC has an excircle beyond A, then its radius RA is given by tanh(RA) ...Missing: derivation | Show results with:derivation<|control11|><|separator|>