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Isogonal conjugates, MP perpendicular to CF

Source: Serbia Additional TST 2012, Problem 3

May 19, 2012
geometrycircumcircleradical axisgeometry proposed

Problem Statement

Let PP and QQ be points inside triangle ABCABC satisfying ∠PAC=∠QAB\angle PAC=\angle QAB and ∠PBC=∠QBA\angle PBC=\angle QBA.
a) Prove that feet of perpendiculars from PP and QQ on the sides of triangle ABCABC are concyclic.
b) Let DD and EE be feet of perpendiculars from PP on the lines BCBC and ACAC and FF foot of perpendicular from QQ on ABAB. Let MM be intersection point of DEDE and ABAB. Prove that MP⊥CFMP\bot CF.
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