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