MathDB

11.8

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(2)

angle chasing, 3 angles 120^o (V Soros Olympiad 1998-99 Round 1 11.8)

Source:

5/25/2024
Inside triangle ABCABC, point PP is taken so that angles ∠ARB=∠BPC=∠CPA=120o\angle ARB= \angle BPC = \angle CPA= 120^o. Lines BPBP and CPCP intersect lines ACAC and ABAB at points MM and KK. It is known that the quadrilateral AMPKAMPK has same areq with the triangle BCPBCP. What is the angle ∠BAC\angle BAC?
geometryangles
area of non convex quad wanted

Source: V Soros Olympiad 1998-99 Round 3 11.8 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/26/2024
Side BCBC of triangle ABCABC is equal to aa and the opposite angle is equal to θ\theta. A straight line passing through the midpoint of BCBC and the center of the inscribed circle intersects lines ABAB and ACAC at points MM and PP, respectively. Find the area of the quadrilateral (non-convex) BMPCBMPC.
geometry