## X(501) (MIQUEL ASSOCIATE OF INCENTER)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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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) = a[a3 - b3 - c3 - bc(a + b + c) + ab(a - b) + ac(a - c)]/(b + c)

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Miquel's theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P∩BC, B' = P∩CA, C' = P∩AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)

X(501) lies on these lines: 1,229    10,662    21,214    35,110    36,58    215,1364    284,942    572,992    595,1326    759,1385

X(501) = isogonal conjugate of X(502)

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