Construction

From a point draw two rays and at an angle .

Let be on such that . Let be the intersection of the angle bisector of the angle and the perpendicular to at . Thus and is a right triangle with hypotenuse .

Draw the line segment of length with between and and on .

Move along until is on ; call this point . Then set .

Let be the reflection of in .

Then satisfies the stated conditions.

Proof

In the triangle , , , but the exterior angle , so .

Triangles and are congruent, so , and .