X(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Trilinears

\(csc(A + π/3) : csc(B + π/3) : csc(C + π/3)\)

\(sec(A - π/6) : sec(B - π/6) : sec(C - π/6)\)

\(for the Fermat point when A> 2π/3 are simply 1:0:0. To represent the Fermat point in the form f(a,b,c) : f(b,c,a) : f(c,a,b), one must use Boolean variables, as shown at Fermat point. If you have The Geometer's Sketchpad, you can view these sketches:\)

Barycentrics

\(a4 - 2(b2 - c2)2 + a2(b2 + c2 + 4*sqrt(3)*Area(ABC)) : :\)

\((SA + Sqrt[3] S) (SB + SC) + 4 SB SC : :\)

Notes

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 X(13). If each of the angles A, B, C is < 2π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.

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.

If you have The Geometer’s Sketchpad, you can view these sketches: Fermat Dynamic 1st isogonic center Kiepert Hyperbola, showing X(13) and X(14) on the hyperbola, with midpoint X(115) Evans Conic, passing through X(13), X(14), X(15), X(16), X(17), X(18), X(3070), X(3071). X(3054), center of the Evans Conic and 19 other triangle centers. If you have GeoGebra, you can view 1st isogonic center.

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

Let NaNbNc, Na’Nb’Nc’ be the outer and inner Napoleon triangles, respectively. Let A’ be the isogonal conjugate of Na’, wrt NaNbNc, and define B’ and C’ cyclically. The lines NaA’, NbB’, NcC’ concur in X(13). (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 = X(13). The locus of P such that AP bisects BPC is the circumcubic given by the barycentric equation c2xy2 - b2xz2 + (a2 - b2 + c2)y2z - (a2 + b2 - c2)yz2 = 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 X(13)X(15) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) + 31/2)|*((E - 8F)S2)1/2. (Kiminari Shinagawa, February 20, 2018)

Let A’B’C’ be the outer Napoleon triangle and A″B″C″ the inner Napoleon triangle. Let A* be the isogonal conjugate, wrt A’B’C’, of A″, and define B* and C* cyclically. The lines A’A*, B’B*, C’C* concur in X(13). (Randy Hutson, December 2, 2017)

Let F be X(13) or X(14). Let L and L’ be lines through F such that the angle between them is π/3; if you have GeoGebra, see Figure 13A. Let LBC = L∩BC, and define LCA and LAB cyclically. Let L’BC = L’∩BC, and define L’CA and L’AB cyclically. The lines LBCL’CA, LCAL’AB, LABL’BC concur. (Dao Thanh Oai, 2014)

Let F be X(13) or X(14). 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; if you have GeoGebra, see Figure 13B. The points A0, B0, C0 are collinear. (Dao Thanh Oai, 2014)

Video showing circular porisim-orbits of X(13), X(14), X(15), and X(16): 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. –> Loci of Centers of Ellipse-Mounted Triangles. (Dan Reznik, August 26, 2020)

If P is a point not on the line X(13)X(14), then the circle {P, X(13), X(14)}} is orthogonal to the orthocentroidal circle. (Peter Moses, April 22, 2021)

Several notable circles pass through X(13) and X(14). 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:

(1116; 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)

(1637; Dao-Moses-Telv circle; 13, 14, 5000, 5001, 6104, 6105, 6106, 6107, 6108, 6109, 6110, 6111, 24007, 24008)

(1640; 13, 14, 10653, 10654, 32618, 32619)

(8371; Hutson-Parry circle, 2, 13, 14, 111, 476, 5466, 5640, 6032, 6792, 7698, 9140, 9159, 11628, 11639, 11640, 13636, 13722, 14846, 14932, 34320)

(9200; 13, 14, 16, 5623, 11586, 30439)

(9201; 13, 14, 15, 5624, 15743, 30440)

(9202; 13, 14, 16, 5623, 11586, 30439)

(14446; 13, 14, 5616, 5668, 6779, 8172, 11600, 38943)

(14447; 13, 14, 5612, 5669, 6780, 8173, 11601, 38944)

(30574; 13, 14, 80,484, 3464, 5540, 5902, 34301, 37718)

(42731; 13, 14, 112, 1141, 1157, 5667, 5890, 6761, 14644, 14651)

(42732; 13, 14, 112, 1141, 1157, 5667, 5890, 6761, 14644, 14651)

(42733; 4, 13, 14, 2132, 2394, 6794, 22265, 34298)

(42734; 13, 14, 616, 621, 5675, 16260, 39133)

(42735; 13, 14, 617, 621, 5674, 16259, 39132)

(42736; 13, 14, 125, 1637, 11657, 14847, 34310)

(42737; 13, 14, 110, 3448, 34306, 34308)

(42738; 13, 14, 98, 11005, 14223, 23969)

(42730; 13, 14, 74, 5627, 5670, 18331)

(42740; Dao-Parry circle of X(1), 1, 13, 14, 79, 5677, 42748, 42749)

Related circles are discussed in trhe preamble just before X(42740).

X(13) lies on the Neuberg cubic and these 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

X(13) is the {X(6),:ref:X(381) <X(381)>}-harmonic conjugate of X(14). For a list of other harmonic conjugates of X(13), click Tables at the top of this page.

X(13) = reflection of X(i) in X(j) for these (i,j): (14,115), (15,396), (99,619), (298,623), (616,618)

X(13) = isogonal conjugate of X(15)

X(13) = isotomic conjugate of X(298)

X(13) = circumcircle-inverse of X(6104)

X(13) = orthocentroidal-circle inverse of X(14)

X(13) = complement of X(616)

X(13) = anticomplement of X(618)

X(13) = cevapoint of X(15) and X(62)

X(13) = X(i)-cross conjugate of X(j) for these (i,j): (15,18), (30,14), (396,2)

X(13) = trilinear pole of line X(395)X(523) (polar of X(470) wrt polar circle)

X(13) = pole wrt polar circle of trilinear polar of X(470)

X(13) = X(48)-isoconjugate (polar conjugate) of X(470)

X(13) = antigonal image of X(14)

X(13) = reflection of X(14) in line X(115)X(125)

X(13) = X(15)-of-4th-Brocard-triangle

X(13) = X(15)-of-orthocentroidal-triangle

X(13) = orthocorrespondent of X(13)

X(13) = homothetic center of outer Napoleon triangle and antipedal triangle of X(13)

X(13) = inner-Napoleon-to-outer-Napoleon similarity image of X(15)

X(13) = outer-Napoleon-isogonal conjugate of X(3)

X(13) = outer-Napoleon-to-inner-Napoleon similarity image of X(14)

X(13) = orthocenter of X(14)X(98)X(2394)

X(13) = X(15)-of-pedal-triangle of X(13)

X(13) = {X(265),:ref:X(1989) <X(1989)>}-harmonic Conjugate of X(14)

X(13) = homothetic center of (equilateral) antipedal triangle of X(13) and triangle formed by circumcenters of BC:ref:X(13) <X(13)>, CA:ref:X(13) <X(13)>, AB:ref:X(13) <X(13)>

X(13) = homothetic center of triangle formed by circumcenters of BC:ref:X(14) <X(14)>, CA:ref:X(14) <X(14)>, AB:ref:X(14) <X(14)> and triangle formed by nine-point centers of BC:ref:X(13) <X(13)>, CA:ref:X(13) <X(13)>, AB:ref:X(13) <X(13)>

X(13) = Cundy-Parry Phi transform of X(17)

X(13) = Cundy-Parry Psi transform of X(61)

X(13) = Kosnita(X(13),:ref:X(1) <X(1)>) point

X(13) = Kosnita(X(13),:ref:X(13) <X(13)>) point

X(13) = Thomson-isogonal conjugate of X(34317)