arc collinear 8 Hint Consider the trianzle formed by thc sid

arc collinear. 8\' Hint: Consider the trianzle formed by thc sidcs (extcnded) A\'B. B\'C. triangle. nanglc. , C\".1 and the 5 transversals of this

Solution

Choose projective coordinates with

C = (1, 0, 0), C\' = (0, 1, 0), P = (0, 0, 1), A = (1, 1, 1).

On the lines AC, AC\', AP, given by x₂=x₃, x₁=x₃, x₂ = x₁, take the points B, Q, b to be

B = (p, 1, 1), Q = (1, q, 1), B\' = (1, 1, r)

for some p, q, r. The three lines PB, CQ, C\'B\' are x₁ = x₂p, x₂ = x₃q, x₃ = x₁r, so they pass through the same point a if and only if rqp = 1. The condition for the three lines CB\', C\'B and PQ x₂ = x₁q, x₁ = x₃p, x₃ = x₂r to pass through the same point R is rpq = 1. So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so pq = qp. Equivalently, P, Q, R are collinear.

Hence Pappus theorem proved