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Necessary and sufficient condition

Source: Chinese TST 2009 3rd quiz P2

March 22, 2009
geometryperpendicular bisectorangle bisectorLaw of Sines

Problem Statement

In convex quadrilateral ABCD ABCD, CB,DA CB,DA are external angle bisectors of ∠DCA,∠CDB \angle DCA,\angle CDB, respectively. Points E,F E,F lie on the rays AC,BD AC,BD respectively such that CEFD CEFD is cyclic quadrilateral. Point P P lie in the plane of quadrilateral ABCD ABCD such that DA,CB DA,CB are external angle bisectors of ∠PDE,∠PCF \angle PDE,\angle PCF respectively. AD AD intersects BC BC at Q. Q. Prove that P P lies on AB AB if and only if Q Q lies on segment EF EF.
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