Theorem 5.9 Lemma

Lemma: Let P₀, P₁, . . . , P_n be points on the upper half of the unit circle where

P_i = (x_i, y_i)

and

x_o < x_n

Then P₀, P₁, . . . , P_n is a partition of the arc on the upper half of the unit circle if and only if x₀ < x₁ < . . . < x_n.

Proof: The proof of this lemma will be broken up into the following sublemmas

Sublemma 1: If

then

Sublemma 2: If (x_i, y_i) is a point in the plane different from the origin, then the line determined by the origin and (x_i, y_i) is given by the equation

y_ix - x_iy = 0

Sublemma 3: If x₀ < x₁ < . . . < x_n and P_i = (x_i, y_i), then P₀, P₁, . . . , P_n form a partition of the arc between P_o and P_n on the upper half of the unit circle.

Sublemma 4: If P₀, P₁, . . . , P_n form a partition of the arc between P₀ and P_n on the upper half of the unit circle where P_i = (x_i, y_i) and x_o < x_n, then x₀ < x₁ < . . . < x_n.

Proof of Theorem 5.9

next theorem (5.10)