X₁₃

1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA', BB', CC' concur in X₁₃. If each of the angles A, B, C is < 2π/3, then X₁₃ is the only center X for which the angles BXC, CXA, AXB are equal, and X₁₃ minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X₁₃ is equilateral.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₃ (1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Equilateral Side Triangles: △BCA', △CAB', △ABC' erected on sidelines with apexes A', B', C'

• Concurrence Lines: A–A', B–B', C–C' concurring at X₁₃

Trilinears: \(\csc(A + \pi /3) : \csc(B + \pi /3) : \csc(C + \pi /3)\)

Barycentrics: \(a^{4} - 2(b^{2} - c^{2})^{2} + a^{2}(b^{2} + c^{2} + 4 \sqrt{3} \operatorname{area}(ABC)) : :\)

Tripolars: \(1/(a^{4}+a^{2} b^{2}-2 b^{4}+a^{2} c^{2}+4 b^{2} c^{2}-2 c^{4}+2 \sqrt{3} a^{2} S) : :\)

Search Value (6, 9, 13): \(0.075945151805\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sec(A - \pi /6) : \sec(B - \pi /6) : \sec(C - \pi /6)\)

• Barycentrics: \((S_{A} + \sqrt{3} S) (S_{B} + S_{C}) + 4 S_{B} S_{C} : :\)

▸ Linear Combinations1 combination

• X₁₃ = \(3^{1/2}(r^{2} + 2rR + s^{2})\) X₁ - \(6r(3^{1/2}R - 2s)\) X₂ - \(2r(3^{1/2}r + 3s)\) X₃ (Peter Moses, April 2, 2013)

X₁₃ lies on the Neuberg cubic.

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_13: The Isogonic & Napoleon Central Lines • T_18: The Apollonian Circles • T_24: Equilateral, Isogonic & Napoleon Triangles

▸ Incident Central Lines21 lines

{2, 16}, {3, 17}, {4, 61}, {5, 18}, {6, 14}, {11, 202}, {15, 30}, {76, 299}, {80, 1251}, {98, 1080}, {99, 303}, {148, 617}, {203, 1478}, {226, 1081}, {262, 383}, {275, 472}, {298, 532}, {484, 1277}, {531, 671}, {533, 621}, {634, 635}

▸ Functional Relations31 relations

• X₁₃ = reflection of X_i in X_j for these (i,j): (14, 115), (15, 396), (99, 619), (298, 623), (616, 618)

• X₁₃ = isogonal conjugate of X₁₅

• X₁₃ = isotomic conjugate of X₂₉₈

• X₁₃ = circumcircle-inverse of X₆₁₀₄

• X₁₃ = orthocentroidal-circle inverse of X₁₄

• X₁₃ = complement of X₆₁₆

• X₁₃ = anticomplement of X₆₁₈

• X₁₃ = cevapoint of X₁₅ and X₆₂

• X₁₃ = X_i-cross conjugate of X_j for these (i,j): (15, 18), (30, 14), (396, 2)

• X₁₃ = trilinear pole of line X₃₉₅X₅₂₃ (polar of X₄₇₀ wrt polar circle)

• X₁₃ = pole wrt polar circle of trilinear polar of X₄₇₀

• X₁₃ = X₄₈-isoconjugate (polar conjugate) of X₄₇₀

• X₁₃ = antigonal image of X₁₄

• X₁₃ = reflection of X₁₄ in line X₁₁₅X₁₂₅

• X₁₃ = X₁₅-of-4th-Brocard-triangle

• X₁₃ = X₁₅-of-orthocentroidal-triangle

• X₁₃ = orthocorrespondent of X₁₃

• X₁₃ = homothetic center of outer Napoleon triangle and antipedal triangle of X₁₃

• X₁₃ = inner-Napoleon-to-outer-Napoleon similarity image of X₁₅

• X₁₃ = outer-Napoleon-isogonal conjugate of X₃

• X₁₃ = outer-Napoleon-to-inner-Napoleon similarity image of X₁₄

• X₁₃ = orthocenter of X₁₄X₉₈X₂₃₉₄

• X₁₃ = X₁₅-of-pedal-triangle of X₁₃

• X₁₃ = {X₂₆₅,X₁₉₈₉}-harmonic Conjugate of X₁₄

• X₁₃ = homothetic center of (equilateral) antipedal triangle of X₁₃ and triangle formed by circumcenters of BCX₁₃, CAX₁₃, ABX₁₃

• X₁₃ = homothetic center of triangle formed by circumcenters of BCX₁₄, CAX₁₄, ABX₁₄ and triangle formed by nine-point centers of BCX₁₃, CAX₁₃, ABX₁₃

• X₁₃ = Cundy-Parry Phi transform of X₁₇

• X₁₃ = Cundy-Parry Psi transform of X₆₁

• X₁₃ = Kosnita(X₁₃,X₁) point

• X₁₃ = Kosnita(X₁₃,X₁₃) point

• X₁₃ = Thomson-isogonal conjugate of X₃₄₃₁₇

Notes

If, however, A> 2π/3, then the Fermat point, defined geometrically as the minimizer of |AX| + |BX| + |CX|, is not the 1st isogonic center (which is defined by the above trilinears). Trilinears for the Fermat point when A> 2π/3 are simply \(1:0:0\). To represent the Fermat point in the form \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), one must use Boolean variables, as shown at Fermat point.

The Evans conic is introduced in Evans, Lawrence S., “A Conic Through Six Triangle Centers,” Forum Geometricorum 2 (2002) 89-92.

Let N_aN_bN_c, N_a'N_b'N_c' be the outer and inner Napoleon triangles, respectively. Let A' be the isogonal conjugate of N_a', wrt N_aN_bN_c, and define B' and C' cyclically. The lines N_aA', N_bB', N_cC' concur in X₁₃. (Randy Hutson, January 29, 2015)

Let P be a point inside triangle ABC such that the line AP bisects angle BPC, and NBP bisets CPA, and CP bisects APB. Then P = X₁₃. The locus of P such that AP bisects BPC is the circumcubic given by the barycentric equation \(c^{2}\)\(xy^{2}\) - \(b^{2}\)\(xz^{2}\) + (\(a^{2}\) - \(b^{2}\) + \(c^{2})\)\(y^{2}\)z - (\(a^{2}\) + \(b^{2}\) - \(c^{2})\)\(yz^{2}\) = 0, and the other two cubics are given cyclically. Bernard Gibert discusses these cubics as K053A, K053B, K053C; see Apollonian strophoids. (Paul Hanna and Peter Moses, August 6, 2017)

The line X₁₃X₁₅ is parallel to the Euler line, and the distance between the two lines is \((S_{B} - S_{C})(S_{C} - S_{A})(S_{A} - S_{B})/|(\cot(\omega ) + 3^{1/2})|*((E - 8F)S^{2})^{1/2}\). (Kiminari Shinagawa, February 20, 2018)

Let Nap_aNap_bNap_c be the outer Napoleon triangle and Nap'_aNap'_bNap'_c the inner Napoleon triangle. Let A* be the isogonal conjugate, wrt Nap_aNap_bNap_c, of Nap'_a, and define B* and C* cyclically. The lines Nap_aA*, Nap_bB*, Nap_cC* concur in X₁₃. (Randy Hutson, December 2, 2017)

Let F be X₁₃ or X₁₄. Let L and L' be lines through F such that the angle between them is π/3. Let L_BC = L∩BC, and define L_CA and L_AB cyclically. Let L'_BC = L'∩BC, and define L'_CA and L'_AB cyclically. The lines L_BCL'_CA, L_CAL'_AB, L_ABL'_BC concur. (Dao Thanh Oai, 2014)

Let F be X₁₃ or X₁₄. Let A₀, B₀, C₀ be points on BC, CA, AB, respectively, such that the directed angles FA₀-to-FC₀ = π/3 and FC₀-to-FB₀ = π/3. The points A₀, B₀, C₀ are collinear. (Dao Thanh Oai, 2014)

Video showing circular porisim-orbits of X₁₃, X₁₄, X₁₅, and X₁₆: 3-Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of Four Triangle Centers. (Dan Reznik, August 9, 2020) See also Loci of Centers of Ellipse-Mounted Triangles. (Dan Reznik, August 26, 2020)

If P is a point not on the line X₁₃X₁₄, then the circle {P, X₁₃, X₁₄}} is orthogonal to the orthocentroidal circle. (Peter Moses, April 22, 2021)

Several notable circles pass through X₁₃ and X₁₄. For each circle listed here, the appearance of (i; [name], [list]) means that the center is X_i, and the points with listed indices lie on the circle:

• (X₁₁₁₆; Lester circle, 3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365)

• (X₁₆₃₇; Dao-Moses-Telv circle; 13, 14, 5000, 5001, 6104, 6105, 6106, 6107, 6108, 6109, 6110, 6111, 24007, 24008)

• (X₁₆₄₀; 13, 14, 10653, 10654, 32618, 32619)

• (X₈₃₇₁; Hutson-Parry circle, 2, 13, 14, 111, 476, 5466, 5640, 6032, 6792, 7698, 9140, 9159, 11628, 11639, 11640, 13636, 13722, 14846, 14932, 34320)

• (X₉₂₀₀; 13, 14, 16, 5623, 11586, 30439)

• (X₉₂₀₁; 13, 14, 15, 5624, 15743, 30440)

• (X₉₂₀₂; 13, 14, 16, 5623, 11586, 30439)

• (X₁₄₄₄₆; 13, 14, 5616, 5668, 6779, 8172, 11600, 38943)

• (X₁₄₄₄₇; 13, 14, 5612, 5669, 6780, 8173, 11601, 38944)

• (X₃₀₅₇₄; 13, 14, 80,484, 3464, 5540, 5902, 34301, 37718)

• (X₄₂₇₃₁; 13, 14, 112, 1141, 1157, 5667, 5890, 6761, 14644, 14651)

• (X₄₂₇₃₂; 13, 14, 112, 1141, 1157, 5667, 5890, 6761, 14644, 14651)

• (X₄₂₇₃₃; 4, 13, 14, 2132, 2394, 6794, 22265, 34298)

• (X₄₂₇₃₄; 13, 14, 616, 621, 5675, 16260, 39133)

• (X₄₂₇₃₅; 13, 14, 617, 621, 5674, 16259, 39132)

• (X₄₂₇₃₆; 13, 14, 125, 1637, 11657, 14847, 34310)

• (X₄₂₇₃₇; 13, 14, 110, 3448, 34306, 34308)

• (X₄₂₇₃₈; 13, 14, 98, 11005, 14223, 23969)

• (X₄₂₇₃₀; 13, 14, 74, 5627, 5670, 18331)

• (X₄₂₇₄₀; Dao-Parry circle of X₁, 1, 13, 14, 79, 5677, 42748, 42749)

Related circles are discussed in trhe preamble just before X₄₂₇₄₀.

X₁₃ is the {X₆,X₃₈₁}-harmonic conjugate of X₁₄.