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Euler line

The Euler line is a straight line in the plane of a triangle that passes through its orthocenter (the intersection of the altitudes), centroid (the intersection of the medians), and circumcenter (the intersection of the perpendicular bisectors of the sides).^[1] This collinearity was discovered in 1765 by the Swiss mathematician Leonhard Euler while investigating properties of triangle centers.^[2] Along the Euler line, the centroid divides the segment joining the orthocenter and circumcenter in the ratio 2:1, with the longer portion extending from the centroid to the orthocenter.^[1] The nine-point center, which is the center of the circle passing through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices, lies at the midpoint between the orthocenter and circumcenter.^[3] In an equilateral triangle, all these points coincide at the center.^[1] The position of the Euler line relative to the triangle varies with its type: it lies entirely within an acute triangle, passes through a vertex in a right triangle (where the circumcenter is at the midpoint of the hypotenuse), and extends outside an obtuse triangle.^[1] This line is perpendicular to the orthic axis^[4] and de Longchamps line^[5] of the triangle, and it connects numerous other notable points, such as the de Longchamps point (the reflection of the orthocenter over the circumcenter).^[6] Euler's discovery marked a significant advancement in Euclidean geometry, inspiring further exploration of triangle centers and their relations.^[2]

Fundamentals

Definition and Key Centers

The Euler line of a triangle is the straight line that passes through the orthocenter, centroid, and circumcenter.^[7] The orthocenter is the point where the altitudes of the triangle intersect.^[8] The centroid is the point where the medians intersect and serves as the center of mass of the triangle, dividing each median in a 2:1 ratio with the longer segment toward the vertex.^[9] The circumcenter is the center of the circumcircle, equidistant from all three vertices of the triangle.^[10] Several other important triangle centers also lie on the Euler line, including the nine-point center and the de Longchamps point. The nine-point center is the midpoint between the orthocenter and the circumcenter, and it serves as the center of 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. The de Longchamps point is the reflection of the orthocenter over the circumcenter.^[11] In an acute triangle, the orthocenter, centroid, and circumcenter all lie inside the triangle, while in an obtuse triangle, the orthocenter and circumcenter lie outside, with the centroid remaining inside.^[7]

Historical Context

The Euler line is named after the Swiss mathematician Leonhard Euler (1707–1783), who first described its key property in 1763 while residing in Berlin and working at the Prussian Academy of Sciences. In his paper Solutio facilis problematum quorumdam geometricorum difficillimorum (E325), written amid the disruptions of the Seven Years' War and published in the Novi Commentarii Academiae Scientiarum Petropolitanae in 1767, Euler proved that the orthocenter, centroid, and circumcenter of any non-equilateral triangle are collinear, with the centroid dividing the segment joining the orthocenter and circumcenter in the ratio 2:1. This result emerged as a byproduct of Euler's investigation into constructing triangles with given angles and side lengths, revealing an unexpected alignment among these fundamental centers.^[2] Earlier explorations of triangle centers laid groundwork for Euler's insight, though none had identified this specific collinearity. For instance, the orthocenter and circumcenter had been studied in Euclidean geometry since antiquity, with properties formalized by figures like Euclid and Apollonius, but their connection to the centroid—a concept rooted in mechanics and known since Archimedes—remained unlinked until Euler's work. Euler's discovery thus marked a pivotal synthesis, demonstrating how these centers, previously treated separately, share a common line in the plane of the triangle.^[2] In the 19th century, the Euler line became a cornerstone of advancing triangle geometry, inspiring synthetic methods that emphasized collinearities and concyclic points without coordinates. Mathematicians such as Karl Wilhelm Feuerbach and August Ferdinand Möbius built upon Euler's ideas, integrating the line into broader theories of triangle centers and influencing the development of projective geometry. By the late 1800s, it was recognized as a unifying element that bridged classical and modern approaches to planar figures.^[12]

Proofs of Existence

Vector-Based Proof

To derive the collinearity of the orthocenter, circumcenter, and centroid on the Euler line using vector geometry, consider a triangle ABC in the plane with position vectors \mathbf{A}, \mathbf{B}, and \mathbf{C} relative to an arbitrary origin.^[13] The centroid \mathbf{G} is the arithmetic mean of the vertex position vectors:

\mathbf{G} = \frac{\mathbf{A} + \mathbf{B} + \mathbf{C}}{3}.

^[13] The circumcenter \mathbf{O} is the unique point equidistant from A, B, and C, located at the intersection of the perpendicular bisectors of the triangle's sides. For the side AB, the perpendicular bisector equation is (\mathbf{P} - \mathbf{M}_{AB}) \cdot (\mathbf{B} - \mathbf{A}) = 0, where \mathbf{M}_{AB} = \frac{\mathbf{A} + \mathbf{B}}{2} is the midpoint and \cdot denotes the dot product; analogous equations hold for the bisectors of BC and CA. Solving this system of linear equations yields \mathbf{O}.^[14] The orthocenter \mathbf{H}, intersection of the altitudes, satisfies Euler's vector formula \mathbf{H} = \mathbf{A} + \mathbf{B} + \mathbf{C} - 2\mathbf{O}.^[13] Substituting the expression for the centroid gives \mathbf{A} + \mathbf{B} + \mathbf{C} = 3\mathbf{G}, so

\mathbf{H} = 3\mathbf{G} - 2\mathbf{O}.

Rearranging yields

\mathbf{H} - \mathbf{O} = 3(\mathbf{G} - \mathbf{O}),

demonstrating that the vector from \mathbf{O} to \mathbf{H} is a scalar multiple (by 3) of the vector from \mathbf{O} to \mathbf{G}, confirming the collinearity of \mathbf{O}, \mathbf{G}, and \mathbf{H}. The centroid divides the segment OH in the ratio OG:GH = 1:2.^[13] To verify that $3\mathbf{G} - 2\mathbf{O} indeed locates the orthocenter, define \mathbf{H}' = 3\mathbf{G} - 2\mathbf{O} and show it lies on each altitude. For the altitude from A to side BC, compute (\mathbf{H}' - \mathbf{A}) \cdot (\mathbf{C} - \mathbf{B}):

\mathbf{H}' - \mathbf{A} = 3\mathbf{G} - 2\mathbf{O} - \mathbf{A} = (\mathbf{A} + \mathbf{B} + \mathbf{C}) - 2\mathbf{O} - \mathbf{A} = \mathbf{B} + \mathbf{C} - 2\mathbf{O}.

Then

(\mathbf{H}' - \mathbf{A}) \cdot (\mathbf{C} - \mathbf{B}) = (\mathbf{B} + \mathbf{C} - 2\mathbf{O}) \cdot (\mathbf{C} - \mathbf{B}) = (\mathbf{C} \cdot \mathbf{C} - \mathbf{C} \cdot \mathbf{B} + \mathbf{B} \cdot \mathbf{C} - \mathbf{B} \cdot \mathbf{B}) - 2\mathbf{O} \cdot (\mathbf{C} - \mathbf{B}) = (|\mathbf{C}|^2 - |\mathbf{B}|^2) - 2\mathbf{O} \cdot (\mathbf{C} - \mathbf{B}).

Since \mathbf{O} is the circumcenter, |\mathbf{O} - \mathbf{A}|^2 = |\mathbf{O} - \mathbf{B}|^2 = |\mathbf{O} - \mathbf{C}|^2, implying |\mathbf{B}|^2 - 2\mathbf{B} \cdot \mathbf{O} = |\mathbf{C}|^2 - 2\mathbf{C} \cdot \mathbf{O}, or |\mathbf{B}|^2 - |\mathbf{C}|^2 = 2\mathbf{O} \cdot (\mathbf{B} - \mathbf{C}). Substituting gives

(|\mathbf{C}|^2 - |\mathbf{B}|^2) - 2\mathbf{O} \cdot (\mathbf{C} - \mathbf{B}) = - (|\mathbf{B}|^2 - |\mathbf{C}|^2) - 2\mathbf{O} \cdot (\mathbf{C} - \mathbf{B}) = -2\mathbf{O} \cdot (\mathbf{B} - \mathbf{C}) - 2\mathbf{O} \cdot (\mathbf{C} - \mathbf{B}) = 0.

Thus, \mathbf{H}' - \mathbf{A} is perpendicular to \mathbf{C} - \mathbf{B}. Symmetric calculations confirm \mathbf{H}' lies on the other altitudes, so \mathbf{H} = \mathbf{H}'.^[13] The collinearity can be verified parametrically: the position vector along the line is \mathbf{P}(t) = \mathbf{O} + t (\mathbf{G} - \mathbf{O}), where t = 0 at \mathbf{O}, t = 1 at \mathbf{G}, and t = 3 at \mathbf{H}.^[13]

Geometric Proof Using Similar Triangles

One elegant geometric proof of the collinearity of the orthocenter H, centroid G, and circumcenter O in a triangle ABC relies on the properties of the medial triangle and homothety.^[15] The medial triangle, denoted DEF where D, E, and F are the midpoints of sides BC, CA, and AB respectively, is similar to \triangle ABC with a scale factor of $1/2.^[16] Moreover, the sides of \triangle DEF are parallel to the sides of \triangle ABC, as each side of the medial triangle connects midpoints and thus is parallel to the third side of the original triangle by the midpoint theorem.^[15] A key observation is that O, the circumcenter of \triangle ABC, serves as the orthocenter of the medial triangle \triangle DEF.^[16] This follows because the altitudes of \triangle DEF are the perpendicular bisectors of \triangle ABC, which concur at O. The centroid G is shared by both triangles, as it is the intersection of the medians of \triangle ABC and lies at the average of the vertices, which coincides for the medial triangle.^[15] To establish alignment, consider a homothety centered at G with ratio -1/2, which maps \triangle ABC to \triangle DEF.^[11] Under this homothety, the orthocenter H of \triangle ABC maps to the orthocenter of \triangle DEF, which is O. Since a homothety preserves lines and maps points along rays from the center, the image of the line through H and G must pass through the image of H, namely O, implying H, G, and O are collinear.^[11] For a direct demonstration using similar triangles, focus on specific constructions involving the medial triangle. Consider the altitude from C to side AB at foot X and the median from C to midpoint F of AB. The line FY, where Y is the foot of the perpendicular from F to a relevant transversal, is parallel to the altitude XC due to the parallel sides of the medial and original triangles.^[16] This parallelism creates alternate interior angles: \angle OFG = \angle ICG, where I denotes H for clarity in the construction. Triangles \triangle FOG and \triangle CIG (with I = H) are similar by SAS, as they share the angle equality, with side ratios CG = 2 \cdot GF and CI = 2 \cdot FO stemming from the centroid dividing medians in a 2:1 ratio and the 2:1 similarity of the triangles. The similarity implies corresponding angles are equal, including those at G, confirming that the line OI (through O, G, I=H) is straight, as the transversal aligns without deviation. A supporting lemma notes that the line from H through G intersects the circumcircle of \triangle ABC at the reflections of H over the sides, but the collinearity follows directly from the transitive alignment along this ray.^[15] This synthetic approach yields the same ratio HG:GO = 2:1 as the vector relation \mathbf{H} = 3\mathbf{G} - 2\mathbf{O}, confirming the positions without coordinates.^[11]

Core Properties

Distances and Ratios Between Centers

The centroid G divides the segment joining the orthocenter H and the circumcenter O in the ratio HG:GO = 2:1, meaning the distance from H to G is twice that from G to O.^[7] This fixed ratio positions G such that OG = \frac{1}{3} OH, where OH denotes the distance between O and H.^[7] The distance OH satisfies the formula

OH^2 = 9R^2 - (a^2 + b^2 + c^2),

where R is the circumradius and a, b, c are the side lengths of the triangle.^[7] This relation provides a direct way to compute the separation between O and H from basic triangle elements. The nine-point center N, which is the center of the nine-point circle passing through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices, lies midway between O and H, so ON:NH = 1:1 and ON = \frac{1}{2} OH.^[7] Consequently, N is positioned between G and H, with the distance GN = \frac{1}{2} GO.^[7] The de Longchamps point L is the reflection of the orthocenter H over the circumcenter O, placing O as the midpoint of segment HL.^[17] Thus, OL = OH, and given the position of G, this implies OL = 3 \, OG in magnitude, with L located on the extension of the Euler line beyond O in the direction opposite to H.^[17] In general, the positions of these centers along the Euler line can be parameterized using vector differences, where the distance d between any two points, such as P and Q, is proportional to the magnitude of their position vectors relative to a reference point like O, scaled by the established ratios.^[7]

Collinearity and Direction

The Euler line of a triangle is defined by the collinear points of its orthocenter H, centroid G, and circumcenter O, establishing a unique direction for the line in any non-equilateral triangle.^[7] This direction can be represented by the vector \overrightarrow{OH} from the circumcenter to the orthocenter or, equivalently, by \overrightarrow{GO} from the centroid to the circumcenter, reflecting the fixed 2:1 ratio in which G divides the segment OH.^[7] The line's orientation is generally arbitrary relative to the triangle's sides, lacking parallelism to any side except in cases of specific symmetry, and it exhibits perpendicularity to certain other triangle lines, such as the de Longchamps line and the orthic axis.^[7] In the degenerate case of an equilateral triangle, the orthocenter, centroid, and circumcenter coincide at a single point, causing the Euler line to collapse into this point rather than extending as a proper line.^[7]^[18] The Euler line is inherently unique for each triangle, as it is the sole line accommodating these key centers, and this configuration remains invariant under similarity transformations, which preserve the positions of H, G, and O relative to one another.^[7] As an infinite line, the Euler line extends beyond the segment OH in both directions, passing through additional notable points derived from reflections of the centers, such as the de Longchamps point, which is the reflection of H over O.^[7]^[11] In limiting configurations, such as those involving the excentral triangle formed by the excenters, the Euler line aligns with lines connecting the incenter and circumcenter, incorporating excentral elements in extended geometric contexts.^[19]

Coordinate Representations

Cartesian Equation in Coordinate Geometry

To derive the Cartesian equation of the Euler line for a triangle in the coordinate plane, consider a general triangle with vertices A(x_1, y_1), B(x_2, y_2), and C(x_3, y_3). The Euler line passes through key centers such as the orthocenter H and the circumcenter O. The coordinates of these points can be computed explicitly, allowing the line to be expressed in standard Cartesian form. The orthocenter H is the intersection point of the triangle's altitudes. To find its coordinates, determine the equations of two altitudes using perpendicular slopes. For instance, the altitude from A to side BC has slope m_{BC} = \frac{y_3 - y_2}{x_3 - x_2}, so the perpendicular slope is -\frac{1}{m_{BC}} = -\frac{x_3 - x_2}{y_3 - y_2}. The equation of this altitude is y - y_1 = \left(-\frac{x_3 - x_2}{y_3 - y_2}\right)(x - x_1). Similarly, the altitude from B to side AC has perpendicular slope -\frac{x_1 - x_3}{y_1 - y_3}, yielding the equation y - y_2 = \left(-\frac{x_1 - x_3}{y_1 - y_3}\right)(x - x_2). Solving this system of two linear equations gives the coordinates (x_H, y_H) of H. The circumcenter O is the intersection of the perpendicular bisectors of the sides and has coordinates

x_O = \frac{(x_1^2 + y_1^2)(y_2 - y_3) + (x_2^2 + y_2^2)(y_3 - y_1) + (x_3^2 + y_3^2)(y_1 - y_2)}{2D},

y_O = \frac{(x_1^2 + y_1^2)(x_3 - x_2) + (x_2^2 + y_2^2)(x_1 - x_3) + (x_3^2 + y_3^2)(x_2 - x_1)}{2D},

where D = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) is twice the signed area of the triangle.^[10] With (x_H, y_H) and (x_O, y_O) known, the Euler line can be written in the two-point form:

\frac{y - y_O}{x - x_O} = \frac{y_H - y_O}{x_H - x_O},

assuming x_H \neq x_O; if vertical, the equation simplifies to x = x_O. This rearranges to the general Cartesian form a x + b y + c = 0, where a = y_O - y_H, b = x_H - x_O, and c = x_O y_H - x_H y_O. For a simplified representation, translate the coordinate system so the centroid G (with coordinates \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)) is at the origin by setting x' = x - G_x and y' = y - G_y. In this position, the vector relation \mathbf{H} = 3\mathbf{G} - 2\mathbf{O} implies \mathbf{H} = -2\mathbf{O} since \mathbf{G} = \mathbf{0}, so the Euler line passes through the origin and is the line spanned by the position vector of O' (the translated circumcenter). Its equation is y' x_{O'} - x' y_{O'} = 0, or in unprimed coordinates, (y - G_y)(x_O - G_x) - (x - G_x)(y_O - G_y) = 0.^[20] As an example, consider the right triangle with vertices A(0,0), B(4,0), and C(0,3). The altitudes from A and C intersect at H(0,0). The circumcenter O is at the midpoint of the hypotenuse BC, so O(2, 1.5). The two-point form gives \frac{y - 1.5}{x - 2} = \frac{0 - 1.5}{0 - 2} = 0.75, or y = 0.75x, which passes through G\left(\frac{4}{3}, 1\right) since $0.75 \cdot \frac{4}{3} = 1.^[10]

Parametric Representation

The parametric representation of the Euler line provides a convenient way to describe points along the line using a scalar parameter t, enabling interpolation and computation of positions relative to key triangle centers such as the circumcenter O, centroid G, nine-point center N, and orthocenter H. This approach is particularly useful in vector-based geometry, where the line is treated as a one-dimensional affine space within the plane of the triangle. In vector notation, a standard parameterization is given by

\mathbf{P}(t) = (1 - t) \mathbf{O} + t \mathbf{H},

where \mathbf{O} and \mathbf{H} are the position vectors of the circumcenter and orthocenter, respectively. This form corresponds to t = 0 at O, t = \frac{1}{3} at G, t = \frac{1}{2} at N, and t = 1 at H. An equivalent form centered at the centroid shifts the origin to G, expressed as

\mathbf{P}(t) = \mathbf{G} + t (\mathbf{H} - \mathbf{O}).

Here, the parameter values are t = -\frac{1}{3} at O, t = 0 at G, t = \frac{1}{6} at N, and t = \frac{2}{3} at H. This centering highlights the centroid's role as dividing the segment OH in the ratio 1:2. The parameterization facilitates the location of additional centers on the Euler line. For instance, the point at t = 2 in the first form, \mathbf{P}(2) = 2\mathbf{H} - \mathbf{O}, aids in computations involving reflections, such as determining the Euler reflection point, the concurrency point of reflections of the Euler line over the triangle's sides.^[21] Using the relation \mathbf{H} = 3\mathbf{G} - 2\mathbf{O}, the direction vector \mathbf{H} - \mathbf{O} = 3(\mathbf{G} - \mathbf{O}) yields a normalized vector form

\mathbf{P}(t) = \mathbf{O} + t \cdot 3(\mathbf{G} - \mathbf{O}),

where t = \frac{1}{3} locates G and t = 1 locates H. This expression underscores the thrice-fold relationship between the vectors from O to G and from O to H. In practice, this parametric form is employed in dynamic geometry software for visualizing the Euler line and animating the movement of centers as triangle vertices vary, supporting interactive exploration of collinearity and ratios.

Applications in Special Triangles

Euler Line in Right Triangles

In a right triangle, the orthocenter coincides with the vertex where the right angle is formed, as the altitudes from the acute vertices are the legs of the triangle, intersecting at this vertex.^[1] The circumcenter is situated at the midpoint of the hypotenuse, serving as the center of the circumcircle with radius equal to half the hypotenuse length.^[1] The centroid, formed by the intersection of the medians and the average of the vertices' coordinates, lies along the median from the right-angled vertex to the midpoint of the hypotenuse, dividing this median in a 2:1 ratio with the longer portion toward the vertex.^[22] The Euler line connects these points, passing from the right-angled vertex (orthocenter) through the centroid to the hypotenuse midpoint (circumcenter).^[22] The standard collinearity ratios hold, with the centroid dividing the orthocenter-circumcenter segment in the ratio 2:1 (orthocenter to centroid being twice the centroid to circumcenter).^[22] The distance between the orthocenter and circumcenter equals the circumradius R, or half the hypotenuse length.^[1] In an isosceles right triangle, the Euler line aligns with the altitude from the right-angled vertex to the hypotenuse, making it perpendicular to the hypotenuse due to the triangle's symmetry.^[22] For a concrete illustration, consider a 3-4-5 right triangle with vertices at A(0,0) (right angle), B(3,0), and C(0,4). The orthocenter is at H = (0,0).^[1] The circumcenter is at O = \left( \frac{3}{2}, 2 \right), the midpoint of the hypotenuse BC.^[1] The centroid is at G = \left( 1, \frac{4}{3} \right), obtained as the average of the coordinates.^[22] These points lie on the line from (0,0) to \left( \frac{3}{2}, 2 \right), with slope \frac{4}{3}, confirming collinearity. The distance OH = \sqrt{ \left( \frac{3}{2} \right)^2 + 2^2 } = \frac{5}{2}, half the hypotenuse of length 5.^[22]

Euler Line in Isosceles Triangles

In an isosceles triangle with equal sides AB = AC and base BC, the Euler line coincides with the median and altitude from the apex A to the midpoint D of the base, forming the axis of symmetry. This alignment stems from the triangle's bilateral symmetry, which positions the orthocenter, centroid, and circumcenter along this single line.^[23]^[22] The centroid divides this altitude in the ratio 2:1, lying at a distance of two-thirds the altitude length from the apex A. The orthocenter and circumcenter also reside on this axis, with their relative positions depending on the apex angle ∠BAC: for ∠BAC < 60°, the circumcenter is nearest to A, followed by the centroid; for ∠BAC > 60°, the orthocenter is nearest to A. The circumradius R, which determines the circumcenter's distance to the vertices, is given by R = \frac{b}{2 \sin A}, where b is the length of the base BC (opposite ∠A) and A = ∠BAC.^[23]^[24] A special case occurs when the base angles are each 45°, making the apex angle 90° and yielding a right-angled isosceles triangle; here, the Euler line remains perpendicular to the base, with the orthocenter at A and the circumcenter at D. This configuration overlaps with Euler line properties in right triangles but underscores the symmetry inherent to isosceles forms.^[23] For visualization purposes, orienting the base BC horizontally positions the Euler line vertically along the y-axis, which simplifies coordinate-based analysis by aligning all key centers on the line x = 0.^[11]

Euler Line in Automedian Triangles

An automedian triangle is defined as a triangle in which the lengths of the medians are proportional to the lengths of the sides, though in a permuted order.^[25] If the sides opposite vertices A, B, and C are a, b, and c respectively, and the medians from these vertices are m_a, m_b, and m_c, then m_a : m_b : m_c = a : c : b (or a cyclic permutation thereof), satisfying a quadratic relation such as a^2 + c^2 = 2b^2.^[26] This proportionality arises from the median length formula m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}, leading to specific side conditions that ensure the medians mirror the sides up to scaling and reordering.^[25] In an automedian triangle, the Euler line—connecting the orthocenter O, centroid G, and circumcenter—exhibits a distinctive perpendicularity to one of the medians. Specifically, assuming the side condition $2a^2 = b^2 + c^2, the Euler line segment OG is perpendicular to the median from vertex A to the midpoint of side BC.^[27] This orthogonality introduces an additional symmetry not present in general triangles, where the Euler line aligns arbitrarily with respect to the medians. The centroid G, as the intersection of the medians, divides the Euler line in the standard 2:1 ratio (with the longer segment toward the orthocenter), but the perpendicular relation alters the geometric interplay between these centers and the cevian framework.^[27] The symmedian point K (also known as the Lemoine point), the concurrency point of the symmedians, bears a special relation to the Euler line in automedian triangles. The line GK is parallel to side BC, reflecting the proportional cevian structure and enhancing the alignment properties of the configuration.^[27] This parallelism underscores the automedian triangle's self-dual nature with respect to its cevians, where the symmedians—reflections of the medians over the angle bisectors—interact symmetrically with the Euler line's direction. Automedian triangles represent a rare class, as the side conditions restrict them to non-isosceles scalene forms satisfying the quadratic relations. An example is the primitive integer-sided automedian triangle with sides 7, 13, and 17 (satisfying $7^2 + 17^2 = 2 \times 13^2), whose medians are in the ratio 13:7:17.^[25] In this configuration, the Euler line's perpendicularity to the median opposite the side of length 7 provides a concrete illustration of the enhanced symmetries.^[27]

Advanced Relations and Constructions

Connection to Inscribed Equilateral Triangles

Napoleon's theorem states that if equilateral triangles are constructed outwardly (or inwardly) on the sides of any triangle ABC, then the centroids of these three equilateral triangles form another equilateral triangle, known as the outer (or inner) Napoleon triangle.^[28] The centroids of the individual equilateral triangles serve as the vertices of the Napoleon triangle.^[29] Remarkably, the centroid of the Napoleon triangle coincides with the centroid G of the original triangle ABC.^[30] Since G lies on the Euler line of ABC, this establishes a direct connection between the Napoleon configuration and the Euler line.^[31] The first and second Napoleon points, denoted X(17) and X(18) in the Encyclopedia of Triangle Centers, are defined as the concurrence points of the lines joining each vertex of ABC to the remote vertex of the equilateral triangle erected on the opposite side (outward for X(17), inward for X(18)).^[32] The line joining these two Napoleon points is termed the Napoleon line.^[33] This Napoleon line intersects the Euler line of ABC at a point that serves as the radical center of the circumcircle of ABC and two other circles associated with the Napoleon configuration.^[33] In certain configurations, such as when ABC is isosceles, the centroids of the equilateral triangles may lie collinearly with points on the Euler line.^[32] Furthermore, the construction of outward equilateral triangles on the sides of ABC provides a geometric method to locate the Fermat-Torricelli point, which minimizes the total distance from a point to the three vertices of ABC (for triangles with all angles less than 120°).^[34] Specifically, the lines connecting each remote vertex of these equilateral triangles to the opposite vertex of ABC concur at the Fermat-Torricelli point.^[34] This point relates to the Euler line through shared incidences with other centers, such as the centroid, in special cases like equilateral triangles where all relevant points coincide.^[31] Historically, while Napoleon's theorem is attributed to Napoleon Bonaparte (though first published by others in 1825), Leonhard Euler contributed foundational work on triangle centers, including the collinearity of key points that define the Euler line, laying groundwork for later explorations of polygonal constructions like those in the Napoleon configuration.^[28]

Systems of Triangles with Concurrent Euler Lines

In triangle geometry, configurations known as systems of triangles with concurrent Euler lines involve multiple related triangles whose individual Euler lines intersect at a common point, revealing deep symmetries and properties. A classic example is the orthocentric system, formed by a reference triangle ABC and its orthocenter H. This system comprises four triangles: ABC (with orthocenter H), ABH (with orthocenter C), BCH (with orthocenter A), and CAH (with orthocenter B). The Euler lines of these four triangles are concurrent at the system's common nine-point center N, which serves as the midpoint between each triangle's orthocenter and circumcenter. This concurrency underscores the shared nine-point circle across the system and is a cornerstone property in advanced concurrency theorems.^[35] The orthic triangle (formed by the feet of the altitudes from ABC) and the tangential triangle (formed by the lines tangent to the circumcircle at A, B, and C) participate in orthocentric systems with ABC and H. In such setups, their Euler lines concur with those of the reference triangle at N, with the trilinear pole of the orthic axis (the line joining the midpoints of the sides of the orthic triangle) playing a key role in establishing perspectivity and shared concurrency points via trilinear polarity. These relations extend to broader orthocentric quartets where the trilinear pole of key lines like the Euler line determines the intersection.^[36] Poristic systems provide another framework, consisting of families of triangles sharing a fixed incircle and circumcircle, parameterized by rotation around the incenter. While individual Euler lines vary, they all pass through the fixed circumcenter O, and extensions to systems sharing both the circumcircle and Euler circle (the nine-point circle) result in all triangles having the identical fixed Euler line, ensuring concurrency along the entire line. Such configurations highlight invariants in poristic families, with centers like the Bevan point X(40) remaining fixed and influencing line intersections.^[37]^[38] A notable example occurs in the complete quadrilateral, defined by four lines in general position intersecting at six points. The four triangles formed by selecting any three of these lines have Euler lines concurrent at a point called the Euler point of the quadrilateral. This concurrency generalizes properties of perspective triangles and appears in configurations like cyclic quadrilaterals divided by diagonals.^[39] These systems find applications in advanced triangle geometry, particularly in proving concurrency theorems such as Dao's theorem (where Euler lines of three triangles formed by a line parallel to a side and vertex projections concur) and extensions to four concurrent Euler lines using tripolar coordinates. Such results facilitate explorations of loci, perspectivities, and invariants in complex configurations.^[40]^[41]

Generalizations to Higher Dimensions

Euler Line in Quadrilaterals

The concept of the Euler line extends to quadrilaterals through the notion of a quasi-Euler line, which connects analogous centers derived from the four triangles formed by the quadrilateral's vertices. For a general quadrilateral ABCD, consider the triangles ABC, BCD, CDA, and DAB; the quasi-orthocenter H is the intersection of the lines joining the orthocenters of opposite triangles (e.g., orthocenters of ABC and CDA), the quasi-circumcenter O is the intersection of the lines joining the circumcenters of opposite triangles (coinciding with the intersection of the diagonals AC and BD), and the centroid G is the intersection of the lines joining the centroids of opposite triangles (also the centroid of the quadrilateral's vertices). These points H, G, and O are collinear on the quasi-Euler line, with G dividing HO in the ratio HG:GO = 2:1, mirroring the property in triangles.^[42] In cyclic quadrilaterals, which possess a circumcircle, the quasi-Euler line aligns with properties of the shared circumcenter and exhibits homothety relations among the centers of the constituent triangles. Specifically, a dilation of factor -1/3 centered at G maps the quadrilateral to the medial quadrilateral formed by connecting the midpoints of its sides, transforming the circumcenter O to another point on the line, while a dilation of factor 3 from O yields the quasi-orthocenter H, satisfying HG = 3 GO. The nine-point analog, the quasi-nine-point center N, lies midway between H and O, with H'N' = N'O' and H'G' = 2 G'O'.^[42]^[43] For tangential quadrilaterals, which admit an incircle tangent to all four sides, an analogous structure emerges as the Nagel line, passing through the incenter I (center of the incircle), the centroid G, and the Nagel point N (the intersection point of lines from vertices to opposite contact points of the incircle). These points are collinear, with G dividing NI in the ratio NG:GI = 3:1, and the Spieker center S (incenter of the medial triangle) as the midpoint of NI. In bicentric quadrilaterals, which are both cyclic and tangential, the Euler and Nagel lines coincide or exhibit enhanced symmetries, preserving collinearity and ratios akin to those in triangles.^[43] Examples illustrate these properties distinctly. In a square, all centers—quasi-orthocenter, centroid, quasi-circumcenter, incenter—coincide at the intersection of the diagonals, degenerating the line to a point, similar to the equilateral triangle case. In a rectangle, including non-square ones, the quasi-orthocenter, centroid, and quasi-circumcenter all coincide at the intersection of the diagonals, degenerating the quasi-Euler line to a point, similar to the square case.^[43] While the quasi-Euler line is defined for any quadrilateral via these constructed points, classical centers like a single orthocenter or circumcenter do not exist in general, limiting the direct analogy to triangles; meaningful collinearity and ratios akin to the triangular Euler line typically require special conditions such as cyclicity or tangentiality.^[42]

Euler Line in Tetrahedrons

In a tetrahedron, the Euler line is defined as the straight line passing through the centroid G, the circumcenter O, and the orthocenter H, provided the latter exists.^[44] The centroid G is the average of the position vectors of the four vertices, serving as the balance point or center of mass.^[45] The circumcenter O is the point equidistant from all four vertices, which is the center of the unique sphere passing through them and exists for any tetrahedron.^[45] The orthocenter H is the point where the four altitudes—from each vertex perpendicular to the opposite face—intersect.^[44] The orthocenter exists only in orthocentric tetrahedrons, where the altitudes are concurrent, a condition equivalent to the sums of the squares of lengths of opposite edges being equal.^[44] Disphenoids, which have four congruent triangular faces and pairs of equal opposite edges, are orthocentric and thus possess an Euler line.^[46] In a regular tetrahedron, the centroid, circumcenter, and orthocenter coincide at a single point, degenerating the Euler line to that point.^[45] On the Euler line of an orthocentric tetrahedron, the centroid G is the midpoint of the segment joining the orthocenter H and the circumcenter O, so the ratio HG:GO = 1:1.^[44] In vector terms, the position of the orthocenter satisfies \mathbf{H} = 2\mathbf{G} - \mathbf{O}, or equivalently \mathbf{G} = \frac{\mathbf{H} + \mathbf{O}}{2}.^[44] This relation holds due to the symmetry in the vector geometry of the altitudes and midplanes in orthocentric systems.^[46] The Euler line in tetrahedrons finds applications in crystal geometry, where orthocentric structures model certain mineral symmetries, and in 3D simulations for computational geometry and finite element analysis.^[46]

Euler Line in Simplicial Polytopes

In an n-dimensional Euclidean space, an n-simplex is orthocentric if its altitudes concur at a single point known as the orthocenter H_n. For such a simplex, the Euler line is defined as the straight line passing through the orthocenter H_n, the centroid (barycenter) G_n, and the circumcenter O_n.^[47] The centroid G_n is the average of the vertices' position vectors, while the circumcenter O_n is the center of the unique hypersphere passing through all vertices.^[47] The collinearity of H_n, G_n, and O_n follows from the affine structure of Euclidean space, where these points are expressible as affine combinations of the vertices. Specifically, in an orthocentric n-simplex, the centroid divides the segment from the orthocenter to the circumcenter in the ratio H_n G_n : G_n O_n = 2 : (n-1).^[47] This relation yields the vector formula

H_n = \frac{n+1}{n-1} G_n - \frac{2}{n-1} O_n

for n > 1, which generalizes the classical triangle case (n=2) where H = 3G - 2O.^[47] Proofs of this collinearity rely on the concurrency of quasi-medians and properties of the Monge point, which lies on the Euler line and satisfies additional ratio conditions such as O_n G_n : G_n M_n = (n-1) : 2.^[47] The existence of the Euler line requires the simplex to be orthocentric, a condition that becomes more restrictive as n increases, though non-equilateral examples exist in dimensions up to at least 4.^[48] The ratios along the line scale linearly with the dimension n, influencing properties like the position of other centers (e.g., the Monge point). For a 4-simplex (pentachoron), the ratio simplifies to H_4 G_4 : G_4 O_4 = 2 : 3, and orthocentric 4-simplices are studied in contexts such as hypersphere intersections and higher-dimensional analogs of triangle centers.^[47]^[48] Beyond single simplices, the Euler line generalizes to simplicial polytopes—convex hulls of simplices—via the circumcenter of mass (CCM), a volume-weighted average of the circumcenters of a triangulation. The line joining the CCM and the polytope's centroid forms a generalized Euler line, preserving affine invariance and extending to orthocentric-like structures. These constructions apply in higher geometry for analyzing discrete dynamical systems and integrable billiards on polytopes, as well as in optimization over simplicial complexes. Extensions to non-Euclidean settings, such as spherical and hyperbolic simplicial polytopes, maintain collinearity properties but require adjusted metrics for centers like the circumcenter.^[49]

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### Summary of Orthocenter Cartesian Coordinate Formula
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