Fortnightly maths prize for 18 July

How far apart should the centres of the two circles be for the three areas A, B, and C to be equal?

three circles

Mugisha Uwiragiye, Sharif Quansah, Dion Miller, and Juan Florez solved this together, with Mugisha making the decisive breakthrough.


Solution

Suppose the radius of each circle is 1 unit.

Let θ be the angle subtended at the centre C of one of the circles by a line XY drawn between the meeting points of the circles.

Area of sector (“slice” of the circle) CXBY = ½θ (if θ is measured in radians)
Area of triangle CXY = ½sinθ (its height, measured from Y onto the base CX, is sinθ)

Therefore the area of the segment (“cut-off” bit) XBY = ½θ−½sinθ

But that segment is half of the middle area B. If area B = area A, then area B = half the area of either circle = π/2

so area of segment XBY=π/4

sinθ=θ−π/2

Solve that numerically, and get θ=2.310

Distance between centres of circles = 2cos(θ/2) = 2cos(1.155) = 0.808.