MathDB

5

Part of 2013 Oral Moscow Geometry Olympiad

Problems(2)

line MH passes through arc midpoint in a 60-45-75 triangle

Source: 2013 Oral Moscow Geometry Olympiad grades 8-9 p5

8/15/2019
In triangle ABC,∠C=60o,∠A=45oABC, \angle C= 60^o, \angle A= 45^o. Let MM be the midpoint of BC,HBC, H be the orthocenter of triangle ABCABC. Prove that line MHMH passes through the midpoint of arc ABAB of the circumcircle of triangle ABCABC.
geometrycircumcircleorthocenter
perpendicularity wanted, heights, midpoints, circumcenter related

Source: 2013 Oral Moscow Geometry Olympiad grades 10-11 p5

8/17/2019
In the acute-angled triangle ABCABC, let APAP and BQBQ be the altitudes, CMCM be the median . Point RR is the midpoint of CMCM. Line PQPQ intersects line ABAB at TT. Prove that OR⊥TCOR \perp TC, where OO is the center of the circumscribed circle of triangle ABCABC.
geometryperpendicularCircumcenter