## X(593) (1ST HATZIPOLAKIS-YIU POINT)

 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.

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

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears            a/(b + c)2 : b/(c + a)2 : c/(a + b)2
Barycentrics    [a/(b + c)]2 : [b/(c + a)]2 : [c/(a + b)]2

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A. Let AB and AC be where O(A) meets lines AB and AC, respectively. Let L(A) be the line joining AB and AC, and define L(B) and L(C) cyclically. Let A' be where L(B) and L(C) meet, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to triangle ABC, and the center of homothety is X(593).

X(593) lies on these lines: 2,261    31,110    36,58    81,757    115,1029    229,1104

X(593) = isogonal conjugate of X(594)

(Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070)

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