MathDB

10.10

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(2)

circle tangent to incircle (V Soros Olympiad 1998-99 Round 1 10.10)

Source:

5/25/2024
A circle inscribed in triangle ABCABC touches BCBC at point KK, MM is the midpoint of the altitude drawn on BCBC. The straight line KMKM intersects the circle inscribed in ABCABC for the second time at point PP. Prove that the circle passing through BB, CC and PP touches the circle inscribed in triangle ABCABC.
geometrytangent circles
fixed point for constant angle

Source: V Soros Olympiad 1998-99 Round 3 10.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/26/2024
A chord ABAB is drawn in a circle. The line ℓ\ell is parallel to ABAB and does not intersect the circle. Let CC be a certain point on the circle (points CC located on one side of ABAB are considered). Lines CACA and CBCB intersect ℓ\ell at points DD and EE. Prove that there exists a fixed point FF of the plane, not lying on line ℓ\ell , such that ∠DFE\angle DFE is constant.
geometryFixed pointfixed