The Farthest Point

Geometry Level 5

Consider the three unit circles above with the first and third ones touching the middle one and all their centers on the xx-axis. From the point (0,1) (0,-1), locate the point on each of the other circles farthest from it. Connect these three points to form a triangle.

The most simplified ratio of the area of triangle to the sum of areas of the three circles can be expressed in this form

abπ(c+d), \dfrac a{b\pi} ( c + \sqrt d) ,

where a,b,ca,b,c and dd are positive integers with a,ba,b coprime and dd square-free. Find a+b+c+da+b+c+d.

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