MathDB

2

Part of 2005 Oral Moscow Geometry Olympiad

Problems(2)

equal angles wanted inside a parallelogram, parallel, angle bisectors

Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p2

10/16/2020
A parallelogram of ABCDABCD is given. Line parallel to ABAB intersects the bisectors of angles AA and CC at points PP and QQ, respectively. Prove that the angles ADPADP and ABQABQ are equal.
(A. Hakobyan)
geometryparallelogramequal anglesangle bisector
2 projections and a chord's midpoint collinear wanted

Source: 2005 Oral Moscow Geometry Olympiad grades 10-11 p2

10/21/2020
On a circle with diameter ABAB, lie points CC and DD. XYXY is the diameter passing through the midpoint KK of the chord CDCD. Point MM is the projection of point XX onto line ACAC, and point NN is the projection of point YY on line BDBD. Prove that points M,NM, N and KK are collinear.
(A. Zaslavsky)
geometrycollineardiameter