IMO Shortlist 2010 problem G7


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June 23, 2013
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Three circular arcs \gamma_1, \gamma_2, and \gamma_3 connect the points A and C. These arcs lie in the same half-plane defined by line AC in such a way that arc \gamma_2 lies between the arcs \gamma_1 and \gamma_3. Point B lies on the segment AC. Let h_1, h_2, and h_3 be three rays starting at B, lying in the same half-plane, h_2 being between h_1 and h_3. For i, j = 1, 2, 3, denote by V_{ij} the point of intersection of h_i and \gamma_j (see the Figure below). Denote by \widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}} the curved quadrilateral, whose sides are the segments V_{ij}V_{il}, V_{kj}V_{kl} and arcs V_{ij}V_{kj} and V_{il}V_{kl}. 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 {{ INVALID LATEX }} are circumscribed, then the curved quadrilateral \widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}} is circumscribed, too.

Proposed by Géza Kós, Hungary
Source: Međunarodna matematička olimpijada, shortlist 2010