MathDB

2

Part of 2012 Poland - Second Round

Problems(2)

Tetrahedron

Source: Polish MO second round 2012

2/19/2012
Prove that for tetrahedron ABCDABCD; vertex DD, center of insphere and centroid of ABCDABCD are collinear iff areas of triangles ABD,BCD,CADABD,BCD,CAD are equal.
geometry3D geometrytetrahedronpyramidgeometry unsolved
Common point

Source: Polish MO second round 2012

2/19/2012
Let ABCABC be a triangle with ∠A=60∘\angle A=60^{\circ} and AB≠ACAB\neq AC, II-incenter, OO-circumcenter. Prove that perpendicular bisector of AIAI, line OIOI and line BCBC have a common point.
geometryincentercircumcirclegeometric transformationreflectiontrigonometryparallelogram