MathDB

9.10

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(2)

tangent circles , given equal angles (V Soros Olympiad 1998-99 Round 1 9.10)

Source:

5/25/2024
On the bisector of angle AA of triangle ABCABC, points DD and FF are taken inside the triangle so that ∠DBC=∠FBA\angle DBC = \angle FBA. Prove that: a) ∠DCB=∠FCA\angle DCB = \angle FCA b) a circle passing through BB and FF and tangent to the segment BCBC is tangle to the circumscribed circle of the triangle ABCABC.
geometrytangent circles
perimeter of triangle wanted (V Soros Olympiad 1998-99 Round 3 9.10)

Source:

5/25/2024
The bisector of angle ∠BAC\angle BAC of triangle ABCABC intersects arc BCBC (not containing point AA) of the circle circumscribed around this triangle at point PP. Segment APAP is divided by side BCBC in ratio kk (counting from vertex AA). Find the perimeter of triangle ABCABC if BC=aBC = a.
ratiogeometryperimeter