Question

(i) The two circles below intersect at P, Q and lines through these points meet the circles at A, B, C, D. The lines AC and BD are not parallel. Prove that if these lines are of equal length, then ABDC is a cyclic quadrilateral.

(ii) In the picture, the circle on the left and right intersect the middle circle at P, Q, R, S; the lines joining them meet the left and right circles at A, B, C, D. Prove that ABDC is a cyclic quadrilateral.

Answer 1

i.

As ABPQ is cyclic

If ∠A = x then ∠BQP = 180-x

If, ∠B = y then ∠APQ = 180-y

As PQCD is cyclic

∠QDC = 180-y (∠CPQ = y)

∠PCD = 180-x (∠DQP = x)

∠A + ∠D = 180° ∠B + ∠C = 180°

∴ ABCD is a cyclic quadrilateral.

ii.

ABCD is a cyclic quadrilateral.