line is the perpendicular bisector of a segment obtained by reflecting twice through two rays 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
Thanks Emmad. As you said, the result follows. The fact you mention combined with the following show that the triangles that need to be shown are congruent in fact are:
the angle Q''Q'Q equals the angle P''P'Q so the arcs they intercept are the same.
Thank you for your answer
Thanks Emmad. As you said, the result follows. The fact you mention combined with the following show that the triangles that need to be shown are congruent in fact are:
the angle Q''Q'Q equals the angle P''P'Q so the arcs they intercept are the same.
Thank you for your answer