X(14) = 2nd ISOGONIC CENTER¶

Trilinears

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

Barycentrics

\(f(a,b,c) : f(b,c,a) : f(c,a,b), where\)

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

Notes

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 - 4*sqrt(3)*Area(ABC)) Barycentrics (SA - Sqrt[3] S) (SB + SC) + 4 SB SC : : 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) : :

X(14) = 31/2(r2 + 2rR + s2)*:ref:X(1) <X(1)> - 6r(31/2R + 2s)*:ref:X(2) <X(2)> - 2r(31/2r - 3s)*:ref:X(3) <X(3)> (Peter Moses, April 2, 2013) Construct the equilateral triangle BA’C having base BC and vertex A’ on the positive 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(14). The antipedal triangle of X(14) is equilateral.

Let NaNbNc, Na’Nb’Nc’ be the outer and inner Napoleon triangles, resp. Let A’ be the isogonal conjugate of Na, wrt Na’Nb’Nc’, and define B’ and C’ cyclically. The lines Na’A’, Nb’B’, Nc’C’ concur in X(14). (Randy Hutson, January 29, 2015)

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(14). (Randy Hutson, December 2, 2017)

The line X(14)X(16) 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)

If you have The Geometer’s Sketchpad, you can view 2nd isogonic center If you have GeoGebra, you can view 2nd isogonic center.

Let O be a point (not necessarily X(3)), let H=ABCDEF be a regular hexagon with center O, and let P be a point. Define 6 triangles T1=ABP, T2=BCP, … , T6=FAP. Claim 1: The points X(14)-of-Ti lie on OP, i=1…6. Next, define 6 reflection triangles T1’ = reflection of T1 in AB, T2’ = reflection of T2 in BC, …, T6’ = reflection of T6 in FA. Claim 2: If P is interior to H, the points X(14)-of-Ti’ lie on a rectangular hyperbola centered at O: Figure. –> Figure. (Dan Reznik, December 10, 2021)

X(14) lies on the Neuberg cubic and these lines: 2,15 3,18 4,62 5,17 6,13 11,203 16,30 76,298 98,383 99,302 148,616 202,1478 226,554 262,1080 275,473 299,533 397,546 484,1276 530,671 532,622 633,636

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

X(14) = reflection of X(i) in X(j) for these (i,j): (13,115), (16,395), (99,618), (299,624), (617,619)

X(14) = isogonal conjugate of X(16)

X(14) = isotomic conjugate of X(299)

X(14) = complement of X(617)

X(14) = anticomplement of X(619)

X(14) = circumcircle-inverse of X(6105)

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

X(14) = cevapoint of X(16) and X(61)

X(14) = X(i)-cross conjugate of X(j) for these (i,j): (16,17), (30,13), (395,2)

X(14) = trilinear pole of line X(396)X(523) (polar of X(471) wrt polar circle)

X(14) = pole wrt polar circle of trilinear polar of X(471)

X(14) = X(48)-isoconjugate (polar conjugate) of X(471)

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

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

X(14) = X(16)-of-4th-Brocard triangle

X(14) = X(16)-of-orthocentroidal-triangle

X(14) = orthocorrespondent of X(14)

X(14) = homothetic center of inner Napoleon triangle and antipedal triangle of X(14)

X(14) = inner-Napoleon-isogonal conjugate of X(3)

X(14) = outer-Napoleon-to-inner-Napoleon similarity image of X(16)

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

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

X(14) = X(16)-of-pedal-triangle of X(14)

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

X(14) = homothetic center of (equilateral) antipedal triangle of X(14) and triangle formed by circumcenters of BC:ref:X(14) <X(14)>, CA:ref:X(14) <X(14)>, AB:ref:X(14) <X(14)>

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

X(14) = Cundy-Parry Phi transform of X(18)

X(14) = Cundy-Parry Psi transform of X(62)

X(14) = Kosnita(X(14),:ref:X(1) <X(1)>) point

X(14) = Kosnita(X(14),:ref:X(14) <X(14)>) point

X(14) = Thomson-isogonal conjugate of X(34318)