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 Collinear and Equal Angles (Posted on 2013-11-09)
Let Γ1 and Γ2 be arbitrary circles that intersect at points P and Q.

Prove or disprove that there exist points M and N such that

(1) M ∈ Γ1\{P,Q},
(2) N ∈ Γ2\{P,Q},
(3) M, N, and P are collinear, and
(4) ∠MQP = ∠NQP.

If they exist, prove or disprove that they can be constructed with
straightedge and compass.

Here is a link to Wolfram MathWorld:
Definition of Set Difference

 Submitted by Bractals Rating: 4.0000 (1 votes) Solution: (Hide) When I thought up the problem I thought that a proof of existence suchas in Harry's post would be given. But, I thought that the construction problem might be impossible (like there exist two rays that trisect an angle, but to construct them with straightedge and compass is impossible). Harry's construction and proof shows that was wrong. Harry's proof is applicable to the first of five configurations: S between rays QP and QN, S on ray QP (with rays QT and QM coinciding), S between rays QM and QP, S or ray M, and S between rays QT and QM. I have proved the construction for the other four and checkout all five with Geometer's Sketchpad. Maybe Harry will give us his thought processes for the construction.

 Subject Author Date See Harry's second post for "The Solution" Bractals 2013-11-17 20:05:39 re(2): Solution Harry 2013-11-17 18:08:26 re: Solution Bractals 2013-11-14 01:26:09 Solution Harry 2013-11-12 18:19:35 Specific case. Almost full solution Jer 2013-11-12 09:53:12

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