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 \(x\)-axis. From the point \( (0,-1)\), 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

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

where \(a,b,c\) and \(d\) are positive integers with \(a,b\) coprime and \(d\) square-free. Find \(a+b+c+d\).


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