Many Cyclic Quads (Posted on 2013-10-19)

	Submitted by Bractals
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The following applies only to cyclic quadrilaterals ABCD whose diagonals AC and BD intersect at point P. A similar argument applies with letters P and Q swapped. Let the centers of Γ1 and Γ2 be O1 and O2 respectively. Let L1 and L2 be points on Γ1 and Γ2 respectively such that PL1 ⊥ PO2 and PL2 ⊥ PO1. Let points M and N satisfy the conditions of Collinear and Equal Angles. Let A be an arbitrary point on the open arc PML1. Ray AP intersects Γ2 again at point C. B is the point on Γ1 such that ∠BQP = ∠PQC. Ray BP intersects Γ2 again at point D. We will show that ABCD is cyclic by showing that side BC subtends equal angles at A and D. If A is in the open arc PM, then    ∠BAC = ∠BAP = 180°-∠BQP = 180°-∠PQC = ∠PDC = ∠BDC else if A is in the open arc ML1, then    ∠BAC = ∠BAP =         ∠BQP = ∠PQC          = ∠PDC = ∠BDC else ABCD is a degenerate cyclic quadrilateral. Since A was chosen arbitrarily in open arc PML1, there exists infinitely many cyclic quadrilaterals. QED

	Subject	Author	Date
	Solution	Harry	2013-10-26 11:42:12