## X(1133) (BURGESS 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           f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = [sin(π/3 - A/3)]/sin(π/3 + A/3)
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Rotate line BC about B away from A through angle B/3, and rotate line BC about C away from A through angle C/3; let A' be the point in which the two rotated lines meet. Define B' and C' cyclically. Let A" be the point of intersection of lines BC' and B'C, and define B" and C" cyclically. The lines AA', BB', CC' concur in X(357), and AA", BB", CC" concur in X(358). The first of these with reference to triangle A'B'C' is X(1133); i.e., X(1133) = X(357)-of-A'B'C'.

A. G. Burgess, "Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides," Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 58-64; page 63.

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