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Viet Nam TST 2014 day 1 problem 3

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March 31, 2014
geometryincentercircumcirclegeometry unsolved

Problem Statement

Let ABCABC be triangle with A<B<CA<B<C and inscribed in a circle (O)(O). On the minor arc ABCABC of (O)(O) and does not contain point AA, choose an arbitrary point DD. Suppose CDCD meets ABAB at EE and BDBD meets ACAC at FF. Let O1O_1 be the incenter of triangle EBDEBD touches with EB,EDEB,ED and tangent to (O)(O). Let O2O_2 be the incenter of triangle FCDFCD, touches with FC,FDFC,FD and tangent to (O)(O). a) MM is a tangency point of O1O_1 with BEBE and NN is a tangency point of O2O_2 with CFCF. Prove that the circle with diameter MNMN has a fixed point. b) A line through MM is parallel to CECE meets ACAC at PP, a line through NN is parallel to BFBF meets ABAB at QQ. Prove that the circumcircles of triangles (AMP),(ANQ)(AMP),(ANQ) are all tangent to a fixed circle.
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