MathDB

5

Part of 2022 Oral Moscow Geometry Olympiad

Problems(2)

locus of intersections of 2 lines, fixed circle and fixed line

Source: 2022 Oral Moscow Geometry Olympiad grades 8-9 p5

4/17/2022
Given a circle and a straight line ABAB passing through its center (points AA and BB are fixed, AA is outside the circle, and BB is inside). Find the locus of the intersection of lines AXAX and BYBY, where XYXY is an arbitrary diameter of the circle.
(A. Akopyan, A. Zaslavsky)
Locusgeometry
circumcircle of CGE tangent to AF wanted, internal tangent circles

Source: 2022 Oral Moscow Geometry Olympiad grades 10-11 p5

4/18/2022
Circle ω\omega is tangent to the interior of the circle Ω\Omega at the point C. Chord ABAB of circle Ω\Omega is tangent to ω\omega. Chords CFCF and BGBG of circle Ω\Omega intersect at point EE lying on ω\omega. Prove that the circumcircle of triangle CGECGE is tangent to straight line AFAF.
(I. Kukharchuk)
tangent circlestangentgeometry