The question is a slightly modified version from Kiselev's Geometry Book 1. Planimetry Exercise 95:
Given an angle $XOY$ and an interior point $A$, find a point $B$ on the side $OX$ and a point $C$ on the side $OY$ such that the perimeter of the triangle $ABC$ is minimal.
The original question limits the angle to be acute, where the question is solved by introducing points symmetric to $A$ with respect to the sides of the angle. My question is whether one can find two distinct such points $B$ and $C$ if the angle is either right or obtuse. Using the same method, when the angle is right the distance is minimized if the two points coincide to the vertex of the angle, and when the angle is obtuse the points become exterior to the angle. However, I could not prove that there are no two distinct points on the sides of the angle that minimize the perimeter. Is there any other method to find such points or can one prove that no such points exist, i.e. the set of possible perimeters does not have the minimum?