MathDB

4

Part of 2015 Oral Moscow Geometry Olympiad

Problems(2)

3 incircles inside a trapezoid, angle bisectors of A,D intersec on side BC

Source: Oral Moscow Geometry Olympiad 2015 grades 8-9 p4

8/8/2019
In trapezoid ABCDABCD, the bisectors of angles AA and DD intersect at point EE lying on the side of BCBC. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base ABAB at the point KK, and two others touch the bisector DEDE at points MM and NN. Prove that BK=MNBK = MN.
geometrytrapezoidangle bisectorincircleequal segments
equal angles wanted, circumcircle, tangents, midpoints given

Source: Oral Moscow Geometry Olympiad 2015 grades 10-11 p4

8/6/2019
In triangle ABCABC, point MM is the midpoint of BC,PBC, P is the intersection point of the tangents at points BB and CC of the circumscribed circle, NN is the midpoint of the segment MPMP. The segment ANAN intersects the circumscribed circle at point QQ. Prove that ∠PMQ=∠MAQ\angle PMQ = \angle MAQ.
geometrycircumcircleequal anglesTangentsmidpoints