MathDB

6

Part of 2016 Mexico National Olmypiad

Problems(1)

Problem 6

Source: Mexican National Olympiad 2016

11/18/2016
Let ABCDABCD a quadrilateral inscribed in a circumference, l1l_1 the parallel to BCBC through AA, and l2l_2 the parallel to ADAD through BB. The line DCDC intersects l1l_1 and l2l_2 at EE and FF, respectively. The perpendicular to l1l_1 through AA intersects BCBC at PP, and the perpendicular to l2l_2 through BB cuts ADAD at QQ. Let Γ1\Gamma_1 and Γ2\Gamma_2 be the circumferences that pass through the vertex of triangles ADEADE and BFCBFC, respectively. Prove that Γ1\Gamma_1 and Γ2\Gamma_2 are tangent to each other if and only if DPDP is perpendicular to CQCQ.
Mexicogeometrytangent circles