MathDB

3

Part of 2017 European Mathematical Cup

Problems(2)

Geometry with a lot of orthogonality

Source: European Mathematical Cup 2017 Junior,P3

12/28/2017
Let ABCABC be an acute triangle. Denote by HH and MM the orthocenter of ABCABC and the midpoint of side BC,BC, respectively. Let YY be a point on ACAC such that YHYH is perpendicular to MHMH and let QQ be a point on BHBH such that QAQA is perpendicular to AM.AM. Let JJ be the second point of intersection of MQMQ and the circle with diameter MY.MY. Prove that HJHJ is perpendicular to AM.AM.
(Steve Dinh)
geometry
EMC 2017 Harder Geo

Source: EMC 2017

12/26/2017
Let ABCABC be a scalene triangle and let its incircle touch sides BCBC, CACA and ABAB at points DD, EE and FF respectively. Let line ADAD intersect this incircle at point XX. Point MM is chosen on the line FXFX so that the quadrilateral AFEMAFEM is cyclic. Let lines AMAM and DEDE intersect at point LL and let QQ be the midpoint of segment AEAE. Point TT is given on the line LQLQ such that the quadrilateral ALDTALDT is cyclic. Let SS be a point such that the quadrilateral TFSATFSA is a parallelogram, and let NN be the second point of intersection of the circumcircle of triangle ASXASX and the line TSTS. Prove that the circumcircles of triangles TANTAN and LSALSA are tangent to each other.
homothetyPascal's Theoremcyclic quadrilateraltangent circlesgeometry