I've been considering this problem as the hand-drawn illustration as shown below.
Unfortunately, I haven't been able to solve it using this approach. My attempt was to demonstrate that two triangles, centered on R, are isosceles triangles, but I have not been successful.
I am currently seeking guidance on where I may have gone wrong (I already read Three lines and three rotations using reflections. and I didn't understand it.)
I am sharing a solution provided by my colleague, Dr. Hosung Joo.
Using the SAS Congruence rule, we have SPR = S'PR and S'QR = S''QR. Consequently, SRS'' is an isosceles triangle.