## X(52) (ORTHOCENTER OF ORTHIC TRIANGLE)

 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 2A cos(B - C) : cos 2B cos(C - A) : cos 2C cos(A - B)
= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A (sec 2B + sec 2C)

Barycentrics    tan A (sec 2B + sec 2C) : tan B (sec 2C + sec 2A) : tan C (sec 2A + sec 2B)

X(52) lies on these lines:
3,6    4,68    5,51    25,155    26,184    30,185    49,195    113,135    114,211    128,134    129,139

X(52) is the {X(5),X(143)}-harmonic conjugate of X(51).

X(52) = reflection of X(I) in X(J) for these (I,J): (3,389), (5,143), (113,1112), (1209,973)
X(52) = isogonal conjugate of X(96)
X(52) = anticomplement of X(1216)
X(52) = inverse-in-Brocard-circle of X(569)
X(52) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,5), (317,467), (324,216)
X(52) = crosspoint of X(4) and X(24)
X(52) = crosssum of X(3) and X(68)

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