Three circles C1, C2, C3 in the plane touch each other (in three different points). Connect the common point of C1 and C2 with the other two common points by straight lines. Show that these lines meet C3 in diametrically opposite points.
Solution
Put ∠O1CA = α, ∠O2BA = β, ∠O3BC = γ. Since the triangles O1CA, O2BA, O3BC are isosceles, we have ∠AO1C = 180o - 2α, ∠AO2B = 180o - 2β, ∠BO3C = 180o - 2γ. These are the three angles of the triangle O1O2O3, so they sum to 180o, and hence α + β + γ = 180o.
Now ∠O3CY = α (opposite angles), ∠O3BX = β (opposite angles), and we defined ∠O3BC = γ. Hence ∠XO3B + ∠BO3C + ∠CO3Y = (180o - 2β) + (180o - 2γ) + (180o - 2α) = 180o, so XO3Y is a straight line. In other words, X and Y are diametrically opposite.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03