In the figure shown, three points have been placed on the edges of triangle ABC, and these points have been connected to form triangle PQR. Segments PR and QR have then been reflected in the lines AB and AC respectively. Then, a new triangle has been created by joining point A with the two reflections of the point R. Explore the connection between the line segments R'R'_1 and AR and explain your result. When the length of AR is minimised what is the relationship between AR and BC..and its significance? Then, if the points where R'R'_1 intersect the triangle ABC are connected with R to form another triangle, what do you notice about the perimeter of this new triangle?
If either Q or P are used to create the reflected points, explore what happens.
What conclusions can you come to that would allow you to solve the original problem?