MathDB

3

Part of 2014 Greece Team Selection Test

Problems(1)

Parallel lines

Source: Greek TST 2014-Pr.3.

6/25/2014
Let ABCABC be an acute,non-isosceles triangle with AB<AC<BCAB<AC<BC.Let D,E,ZD,E,Z be the midpoints of BC,AC,ABBC,AC,AB respectively and segments BK,CLBK,CL are altitudes.In the extension of DZDZ we take a point MM such that the parallel from MM to KLKL crosses the extensions of CA,BA,DECA,BA,DE at S,T,NS,T,N respectively (we extend CACA to AA-side and BABA to AA-side and DEDE to EE-side).If the circumcirle (c1)(c_{1}) of △MBD\triangle{MBD} crosses the line DNDN at RR and the circumcirle (c2)(c_{2}) of △NCD\triangle{NCD} crosses the line DMDM at PP prove that ST∥PRST\parallel PR.
geometrycircumcircletrapezoidgeometry proposed