MathDB

3

Part of 2009 Iran MO (3rd Round)

Problems(2)

Homothetic Triangles

Source: Iran 3rd round 2009 - final exam problem 3

1/2/2015
An arbitary triangle is partitioned to some triangles homothetic with itself. The ratio of homothety of the triangles can be positive or negative. Prove that sum of all homothety ratios equals to 11.
Time allowed for this problem was 45 minutes.
ratiogeometrygeometric transformationhomothetygeometry unsolved
Iran(3rd round)2009

Source: Problem 3 Geometry

9/13/2009
3-There is given a trapezoid ABCD ABCD in the plane with BC∥AD BC\parallel{}AD.We know that the angle bisectors of the angles of the trapezoid are concurrent at O O.Let T T be the intersection of the diagonals AC,BD AC,BD.Let Q Q be on CD CD such that \angle OQD \equal{} 90^\circ.Prove that if the circumcircle of the triangle OTQ OTQ intersects CD CD again at P P then TP∥AD TP\parallel{}AD.
geometrytrapezoidcircumcirclegeometry unsolved