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3a

Part of 2016 Abels Math Contest (Norwegian MO) Final

Problems(1)

X, Y,C' lie on a line iff AC = BC, 3 circles, 3 common tangents, 2 right angles

Source: Norwegian Mathematical Olympiad 2016 - Abel Competition p3a

9/3/2019
Three circles SA,SBS_A, S_B, and SCS_C in the plane with centers in A,BA, B, and CC, respectively, are mutually tangential on the outside. The touchpoint between SAS_A and SBS_B we call C′C', the one SAS_A between SCS_C we call B′B', and the one between SBS_B and SCS_C we call A′A'. The common tangent between SAS_A and SCS_C (passing through B') we call ℓB\ell_B, and the common tangent between SBS_B and SCS_C (passing through A′A') we call ℓA\ell_A. The intersection point of ℓA\ell_A and ℓB\ell_B is called XX. The point YY is located so that ∠XBY\angle XBY and ∠YAX\angle YAX are both right angles. Show that the points X,YX, Y, and C′C' lie on a line if and only if AC=BCAC = BC.
collinearcirclescommon tangentsgeometry