IMO 1985 SL 16

Problem

5b.(BEL 2) If possible, construct an equilateral triangle whose three vertices are on three given circles.

Solution

Let the three circles be \alpha(A, a), \beta(B, b), and \gamma(C, c), and assume c \leqa, b. We denote by RX,ϕ the rotation around X through an angle ϕ. Let PQR be an equilateral triangle, say of positive orientation (the case of negatively oriented \trianglePQR is analogous), with P \in\alpha, Q \in\beta, and R \in\gamma. Then Q = RP,−60◦(R) \inRP,−60◦(\gamma) \cap\beta. Since the center of RP,−60◦(\gamma) is RP,−60◦(C) = RC,60◦(P) and it lies on RC,60◦(\alpha), the union of circles RP,−60◦(\gamma) as P varies on \alpha is the annulus U with center A′ = RC,60◦(A) and radii a −c and a + c. Hence there is a solution if and only if U \cap\beta is nonempty.