Consider a circle O1 with radius R and a point A outside the circle. It is known that ∠BAC=60∘, where AB and AC are tangent to O1. We construct infinitely many circles Oi (i=1,2,…) such that for i>1, Oi is tangent to Oi−1 and Oi+1, that they share the same tangent lines AB and AC with respect to A, and that none of the Oi are larger than O1. Find the total area of these circles. I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.
Problem
Source: Taiwan 1st TST, 1st independent study, question 1
Tags: geometry, trigonometry, ratio, geometric sequence, geometry proposed