The quadrilateral ABCD is such that it circumscribes a circle. The circle inscribed in the triangle ABC touches AC at X and the circleinscribed in the triangle ADC touches AC at Y. Find the value of XY/AC.

Because quadrilateral ABCD circumscribes the circle, the tangent segment theorem gives AB plus CD equals AD plus BC. Let the circle inscribed in triangle ABC touch AC at X, and let the tangency lengths from A and C be AX = s1 and CX = s2. For the circle inscribed in triangle ADC touching AC at Y, the tangency length from A is AY = s1 and from C is CY = s2 because AB plus CD equals AD plus BC. This means X and Y are the exact same point on AC, so the length XY is 0. Therefore, the ratio XY divided by AC is 0.

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