MathDB

6

Part of 2008 Oral Moscow Geometry Olympiad

Problems(2)

criterion for a convex hexagon with parallel opposite sides to be cyclic

Source: 2008 Oral Moscow Geometry Olympiad grades 8-9 p6

10/13/2020
Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle.
(A. Zaslavsky)
geometryhexagonparallelCyclic
PQ passes through circumcenter, similar triangles of projections of 2 points,

Source: 2008 Oral Moscow Geometry Olympiad grades 10-11 p6

10/14/2020
Given a triangle ABCABC and points PP and QQ. It is known that the triangles formed by the projections PP and QQ on the sides of ABCABC are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line PQPQ passes through the center of the circumscribed circle of triangle ABCABC.
(A. Zaslavsky)
geometrysimilar trianglesCircumcentercollinear