MathDB

2

Part of 2011 Polish MO Finals

Problems(2)

Circumcenters coincide

Source: LXII Polish Olympiad 2011, Problem 2

5/24/2011
The incircle of triangle ABCABC is tangent to BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. Consider the triangle formed by the line joining the midpoints of AE,AFAE,AF, the line joining the midpoints of BF,BDBF,BD, and the line joining the midpoints of CD,CECD,CE. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle ABCABC.
geometrycircumcircleincentergeometric transformationdilationperpendicular bisectorpower of a point
Prove that PQ=2HP

Source: LXII Polish Olympiad 2011, Problem 5

4/20/2011
In a tetrahedron ABCDABCD, the four altitudes are concurrent at HH. The line DHDH intersects the plane ABCABC at PP and the circumsphere of ABCDABCD at Q≠DQ\neq D. Prove that PQ=2HPPQ=2HP.
geometry3D geometrytetrahedroncircumcirclegeometry proposed