## X(2089) (3RD MID-ARC 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.

 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 B/2 + cos C/2 - cos A/2) sec A/2 : (cos C/2 + cos A/2 - cos B/2) sec B/2 : (cos A/2 + cos B/2 - cos C/2) sec B/2
= 1/(1 - sin A/2) : 1/(1 - sin B/2) : 1/(1 - sin C/2)

Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A', B', C' be the first points of intersection of the angle bisectors of triangle ABC with its incircle. Let A" B" C" be the triangle formed by the lines tangent to the incircle at A', B', C'. Then A"B"C" is perspective to the intouch triangle of ABC, and the perspector is X(2089). (Darij Grinberg, Hyacinthos #8072, 10/01/03)

X(2089) lies on these lines: 1,167    2,178    7,1488

X(2089) = X(7)-Ceva conjugate of X(174)
X(2089) = X(173)-cross conjugate of X(174)

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