MathDB

2

Part of 2007 Korea National Olympiad

Problems(2)

convex quadrilateral

Source: 2007 Korean MO, 2nd Round, A.M.

8/18/2007
A1B1B2A2 A_{1}B_{1}B_{2}A_{2} is a convex quadrilateral, and A1B1≠A2B2 A_{1}B_{1}\neq A_{2}B_{2}. Show that there exists a point M M such that \frac{A_{1}B_{1}}{A_{2}B_{2}}\equal{}\frac{MA_{1}}{MA_{2}}\equal{}\frac{MB_{1}}{MB_{2}}
geometry unsolvedgeometry
triangle is not isosceles

Source: 2007 Korean MO, 2nd Round, P.M.

8/18/2007
ABC ABC is a triangle which is not isosceles. Let the circumcenter and orthocenter of ABC ABC be O O, H H, respectively, and the altitudes of ABC ABC be AD AD, BC BC, CF CF. Let K≠A K\neq A be the intersection of AD AD and circumcircle of triangle ABC ABC, L L be the intersection of OK OK and BC BC, M M be the midpoint of BC BC, P P be the intersection of AM AM and the line that passes L L and perpendicular to BC BC, Q Q be the intersection of AD AD and the line that passes P P and parallel to MH MH, R R be the intersection of line EQ EQ and AB AB, S S be the intersection of FD FD and BE BE. If OL \equal{} KL, then prove that two lines OH OH and RS RS are orthogonal.
geometrycircumcirclegeometry unsolved