## X(511) (ISOGONAL CONJUGATE OF X(98))

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

 The JRE (Java Runtime Environment) is not enabled in your browser!

This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           cos(A + ω) : cos(B + ω) : cos(C + ω)

= sin A - sin(A + 2ω) : sin B - sin(B + 2ω) : sin C - sin(C + 2ω)
= cos A + cos(A + 2ω) : cos B + cos(B + 2ω) : cos C + cos(C + 2ω) (cf. X(39))
= a(a2b2 + a2c2 - b4 - c4) : b(b2c2 + b2a2 - c4 - a4) : c(c2a2 + c2b2 - a4 - b4)       (M. Iliev, 5/13/07)

Barycentrics    sin A cos(A + ω) : sin B cos(B + ω) : sin C cos(C + ω)

As the isogonal conjugate of a point on the circumcircle, X(511) lies on the line at infinity.

X(511) lies on these lines:
1,256    2,51    3,6    4,69 5,141    20,185    22,184    23,110    24,1092    25,394    26,206    30,512    40,1045    55,611    56,613    66,68    67,265    74,691    98,385    107,450    111,352    114,325    125,858    140,143    155,159    171,181    186,249    287,401    298,1080    299,383    343,427    381,599    549,597

X(511) = orthopoint of X(512)
X(511) = isogonal conjugate of X(98)
X(511) = isotomic conjugate of X(290)
X(511) = anticomplementary conjugate of X(147)
X(511) = complementary conjugate of X(114)
X(511) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,114), (290,2), (297,232)
X(511) = cevapoint of X(385) and X(401)
X(511) = X(I)-cross conjugate of X(J) for these (I,J): (4,114), (290,2), (297,232)
X(511) = crosspoint of X(I) and X(J) for these (I,J): (2,290), (297,325)
X(511) = crosssum of X(I) and X(J) for these (I,J): (2,385), (6,237), (11,659), (523,868)
X(511) = crossdifference of any two points on line X(6)X(523)
X(511) = X(3)-Hirst inverse of X(6)
X(511) = X(I)-line conjugate of X(J) for these (I,J): (3,6), (30,523)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.