MathDB

1

Part of 2011 Oral Moscow Geometry Olympiad

Problems(2)

MK is equal and perpendicular to the diagonal of the rectangle ABCD

Source: 2011 Oral Moscow Geometry Olympiad grades 8-9 p1

4/17/2020
The bisector of angle BB and the bisector of external angle DD of rectangle ABCDABCD intersect side ADAD and line ABAB at points MM and KK, respectively. Prove that the segment MKMK is equal and perpendicular to the diagonal of the rectangle.
geometryrectangleequal segmentsperpendicular
AB is tangent to the cicrumcircle of DEC'

Source: 2011 Oral Moscow Geometry Olympiad grades 10-11 p1

4/7/2020
ADAD and BEBE are the altitudes of the triangle ABCABC. It turned out that the point C′C', symmetric to the vertex CC wrt to the midpoint of the segment DEDE, lies on the side ABAB. Prove that ABAB is tangent to the circle circumscribed around the triangle DEC′DEC'.
geometrycircumcircletangentsymmetry