IMO 2010 Shortlist G7
Three circular arcs γ1, γ2, and γ3 connect the points A and C. These arcs lie in the same half-plane defined by line AC ...
Category: Geometry
Problem
Three circular arcs γ1, γ2, and γ3 connect the points A and C. These arcs lie in the same half-plane defined by line AC in such a way that arc γ2 lies between the arcs γ1 and γ3. Point B lies on the segment AC. Let h1, h2, and h3 be three rays starting at B, lying in the same half-plane, h2 being between h1 and h3. For i,j 1,2,3, denote by Vij the point of intersection of hi and γj (see the Figure below). Denote by Ǒ VijVkj Ǒ VkℓViℓ the curved quadrilateral, whose sides are the segments VijViℓ, VkjVkℓ and arcs VijVkj and ViℓVkℓ. We say that this quadrilateral is circumscribed if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals Ǒ V11V21 Ǒ V22V12, Ǒ V12V22 Ǒ V23V13, Ǒ V21V31 Ǒ V32V22 are circum- scribed, then the curved quadrilateral Ǒ V22V32 Ǒ V33V23 is circumscribed, too. A C h3 h2 h1 V13 V33 V12 V11 V32 B V22 γ3 V23 γ2 γ1 V21 V31 Fig. 1 (Hungary)