MathDB

3

Part of 2023 Francophone Mathematical Olympiad

Problems(2)

2 circles tangent to a third one wanted, one more circle given

Source: P3 Francophone Math Olympiad Junior 2023

5/2/2023
Let Γ\Gamma and Γ′\Gamma' be two circles with centres OO and O′O', such that OO belongs to Γ′\Gamma'. Let MM be a point on Γ′\Gamma', outside of Γ\Gamma. The tangents to Γ\Gamma that go through MM touch Γ\Gamma in two points AA and BB, and cross Γ′\Gamma' again in two points CC and DD. Finally, let EE be the crossing point of the lines ABAB and CDCD. Prove that the circumcircles of the triangles CEO′CEO' and DEO′DEO' are tangent to Γ′\Gamma'.
geometrycircumcircletangent circles
2 circles tangent to a third one , IMO SL 2020 variant

Source: P3 Francophone Math Olympiad Senior 2023

5/2/2023
Let ABCDABCD be a convex quadrilateral, with ∡ABC>90∘\measuredangle ABC > 90^\circ, ∡CDA>90∘\measuredangle CDA > 90^\circ and ∡DAB=∡BCD\measuredangle DAB = \measuredangle BCD. Let EE, FF and GG be the reflections of AA with respect to the lines BCBC, CDCD and DBDB. Finally, let the line BDBD meet the line segment AEAE at a point KK, and the line segment AFAF at a point LL. Prove that the circumcircles of the triangles BEKBEK and DFLDFL are tangent to each other at GG.
geometrytangent circles