## X(1315) (4th MOSES INTERSECTION)

 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            bcf(a,b,c) : caf(b,c,a): abf(c,a,b), where f(a,b,c) is the 1st barycentric given below

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [d2 + (4r - R)R - sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA,
where Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r + R)]2,
d = distance between X(3) and X(4),
SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically (Peter J. C. Moses, 3/2003)

X(1315) is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).

X(1315) lies on this line: 2,3

X(1315) lies on the incircle

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