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Three circles.

Source: IMO Shortlist 1992, Problem 7

October 27, 2004
geometryincenterangle bisectorcirclesIMO ShortlistIMO LonglistCasey s theorem

Problem Statement

Two circles Ω1 \Omega_{1} and Ω2 \Omega_{2} are externally tangent to each other at a point I I, and both of these circles are tangent to a third circle Ω \Omega which encloses the two circles Ω1 \Omega_{1} and Ω2 \Omega_{2}. The common tangent to the two circles Ω1 \Omega_{1} and Ω2 \Omega_{2} at the point I I meets the circle Ω \Omega at a point A A. One common tangent to the circles Ω1 \Omega_{1} and Ω2 \Omega_{2} which doesn't pass through I I meets the circle Ω \Omega at the points B B and C C such that the points A A and I I lie on the same side of the line BC BC. Prove that the point I I is the incenter of triangle ABC ABC. Alternative formulation. Two circles touch externally at a point I I. The two circles lie inside a large circle and both touch it. The chord BC BC of the large circle touches both smaller circles (not at I I). The common tangent to the two smaller circles at the point I I meets the large circle at a point A A, where the points A A and I I are on the same side of the chord BC BC. Show that the point I I is the incenter of triangle ABC ABC.
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