MathDB

7

Part of 2008 Sharygin Geometry Olympiad

Problems(4)

Find the circumradius of triangle

Source: Sharygin contest. The final raund. 2008. Grade 9. Second day. Problem 7

8/31/2008
(A.Zaslavsky) The circumradius of triangle ABC ABC is equal to R R. Another circle with the same radius passes through the orthocenter H H of this triangle and intersect its circumcirle in points X X, Y Y. Point Z Z is the fourth vertex of parallelogram CXZY CXZY. Find the circumradius of triangle ABZ ABZ.
geometrycircumcircleparallelogramgeometric transformationreflectiongeometry unsolved
Find the ratio of angles

Source: Sharygin contest. The final raund. 2008. Grade 8. Second day. Problem 7

8/31/2008
(F.Nilov) Given isosceles triangle ABC ABC with base AC AC and \angle B \equal{} \alpha. The arc AC AC constructed outside the triangle has angular measure equal to β \beta. Two lines passing through B B divide the segment and the arc AC AC into three equal parts. Find the ratio α/β \alpha / \beta.
ratiogeometry unsolvedgeometry
Circle with center on another circle

Source: Sharygin contest 2008. The correspondence round. Problem 7

9/3/2008
(A.Zaslavsky, 8--9) Given a circle and a point O O on it. Another circle with center O O meets the first one at points P P and Q Q. The point C C lies on the first circle, and the lines CP CP, CQ CQ meet the second circle for the second time at points A A and B B. Prove that AB\equal{}PQ.
geometry proposedgeometry
Arcs on two medians

Source: Sharygin contest. The final raund. 2008. Grade 10. Second day. Problem 7

8/31/2008
(F.Nilov) Two arcs with equal angular measure are constructed on the medians AA′ AA' and BB′ BB' of triangle ABC ABC towards vertex C C. Prove that the common chord of the respective circles passes through C C.
geometrypower of a pointradical axisgeometry unsolved