MathDB

4

Part of 2019 Oral Moscow Geometry Olympiad

Problems(2)

tangent circumcircles

Source: 2019 Oral Moscow Geometry Olympiad grades 8-9 p4

5/21/2019
The perpendicular bisector of the bisector BLBL of the triangle ABCABC intersects the bisectors of its external angles AA and CC at points PP and QQ, respectively. Prove that the circle circumscribed around the triangle PBQPBQ is tangent to the circle circumscribed around the triangle ABCABC.
geometryperpendicular bisectortangent circlescircumcircle
angle between touchpoints of excircles of a right triangle

Source: 2019 Oral Moscow Geometry Olympiad grades 10-11 p4

5/22/2019
Given a right triangle ABCABC (∠C=90o\angle C=90^o). The CC-excircle touches the hypotenuse ABAB at point C1,A1C_1, A_1 is the touchpoint of BB-excircle with line BC,B1BC, B_1 is the touchpoint of AA-excircle with line ACAC. Find the angle ∠A1C1B1\angle A_1C_1B_1.
geometryangleright triangleexcircleexcircles