An important principle in geometry involves the question of whether, in the figure for Pro...

An important principle in geometry involves the question of whether, in the figure for Problem 20, m∠AC’B can be made smaller than a preassigned constant by moving C’ out far enough on ray BC. This seems intuitively clear. To prove it, follow this construction (discovered by Legendre): Let ∈ > 0 be the given constant. We must show that for some point C’ on line ℓ, m∠AC'B<∈. Locate points C1, C2, C3, .. . , Cn on ray such that B-C1-C2 and C1C2 = AC1 B-C2-C3 and C2C3 = AC2, and so on. Show that x2 ≤ x1/2, x3 ≤ x2/2 ≤ x1/4, . . . , and, in general, xn+ 1 ≤ x1/2". (See previous problem.) Now finish the argument to show that for some n, m∠ACnB<∈.

*Used in the proof of Theorem 1, Section 6.3.