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VMO 2018 P2

Source: Vietnam Mo 2018 1st day 2nd problem

January 11, 2018
geometry

Problem Statement

We have a scalene acute triangle ABCABC (triangle with no two equal sides) and a point DD on side BCBC. Pick a point EE on side ABAB and a point FF on side ACAC such that ∠DEB=∠DFC\angle DEB=\angle DFC. Lines DF, DEDF,\, DE intersect AB, ACAB,\, AC at points M, NM,\, N, respectively. Denote (I1), (I2)(I_1),\, (I_2) by the circumcircles of triangles DEM, DFNDEM,\, DFN in that order. The circle (J1)(J_1) touches (I1)(I_1) internally at DD and touches ABAB at KK, circle (J2)(J_2) touches (I2)(I_2) internally at DD and touches ACAC at HH. PP is the intersection of (I1), (I2)(I_1),\, (I_2) different from DD. QQ is the intersection of (J1), (J2)(J_1),\, (J_2) different from DD. a. Prove that all points D, P, QD,\, P,\, Q lie on the same line. b. The circumcircles of triangles AEF, AHKAEF,\, AHK intersect at A, GA,\, G. (AEF)(AEF) also cut AQAQ at A, LA,\, L. Prove that the tangent at DD of (DQG)(DQG) cuts EFEF at a point on (DLG)(DLG).
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