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Collinear and Equal Angles (Posted on 2013-11-09) Difficulty: 5 of 5
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 such
as 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.
  

Comments: ( You must be logged in to post comments.)
  Subject Author Date
See Harry's second post for "The Solution"Bractals2013-11-17 20:05:39
re(2): SolutionHarry2013-11-17 18:08:26
re: SolutionBractals2013-11-14 01:26:09
SolutionSolutionHarry2013-11-12 18:19:35
Some ThoughtsSpecific case. Almost full solutionJer2013-11-12 09:53:12
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