# Circumcenter of triangle formed by point reflected in the lines y=x and y=-x

The point $$Q$$ is the image of the point $$P(1, 5)$$ about the line $$y = x$$ and $$R$$ is the image of the point $$Q$$ about the line $$y=−x$$. The circumcenter of the $$ΔPQ$$R is-

Answer given is $$(0,0)$$

My attempt:

Replection of $$(1,5)$$ in $$y=x$$ is $$(5,1)$$ and in $$y=-x$$ is $$(-5,-1)$$. Now I don't see why origin is the circumcenter.

Let $$O = (0,0)$$ then after reflection across $$y=x$$ $$O$$ goes to it self and the same is true after a reflection across $$y=-x$$. So $$OP = OQ$$ and $$OP = OR$$. So $$O$$ is on the same distance from $$P,Q$$ and $$R$$ so it must be a circumcenter of triangle $$PQR$$.
• Why should it be? The line joining $(-1,-5)$ and $(1,5)$ is clearly not perpendicular to $y=-x$ . – tatan Apr 16 '19 at 14:24
Let $$Q$$ be the image of the first reflection and $$R$$ the image of the second. Since the two lines of reflection are perpendicular, $$\angle{QPR}$$ is a right angle, so $$QR$$ must be a diameter of the circle. The midpoint of this segment is the origin.