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Given two rays, L and M, with common origin O, and a point Q inside the acute angle formed by the rays, reflect Q across L to obtain Q' and then reflect Q' across M to obtain Q''. Similarly, reflect Q across M to obtain P' and then reflect P' across L to obtain P''. Show that the line through O and Q is the perpendicular bisector of the segment joining P'' and Q''. How do you do this? It is easy to show that Q and all reflected points lie on the circle with center O that passes through Q. I thought of using reflection across the line through O and Q and then use a symmetry argument but couldn't see just how to implement this idea. I would really appreciate any help. Thanks

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    It would help some of us if you could provide a picture.2012-09-18
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    I'm sorry I didn't provide a picture but I don't know how to do this using a programming language and as a new user I am not allowed to post images2012-09-18
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    @Emmad: I’ve taken the liberty of adding a diagram.2012-09-18
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    @Thanks Brian, this is helpful.2012-09-18
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    If P" and Q" lie on the premituer of a circle with O at the center, then if you draw 2 lines, OP" and OQ" these 2 lines would be equal (each is a radius of the same circle). From this the proof follows.2012-09-19

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