MathDB

3

Part of 2021 Oral Moscow Geometry Olympiad

Problems(2)

Quadrilateral with two equal angles, perpendicularity wanted

Source: 2021 Oral Moscow Geometry Olympiad grades 8-9 p3

5/17/2021
ABCDABCD is a convex quadrilateral such that ∠A=∠C<90∘\angle A = \angle C < 90^{\circ} and ∠ABD=90∘\angle ABD = 90^{\circ}. MM is the midpoint of ACAC. Prove that MBMB is perpendicular to CDCD.
geometryperpendicular lines
A moves on major arc BC and some point is on the same distance from BC

Source: 2021 Oral Moscow Geometry Olympiad grades 10-11 p3

5/17/2021
Circle (O)(O) and its chord BCBC are given. Point AA moves on the major arc BCBC. ALAL is the angle bisector in a triangle ABCABC. Show that the disctance from the circumcenter of triangle AOLAOL to the line BCBC does not depend on the position of point AA.
geometryangle bisectorcircumcircle