Construction

Draw the line segment of length , and let be on the perpendicular to at such that .

Step 1: Construct the inscribed square of the triangle.

Then satisfies the conditions.

Verification

Let the point be the fourth point of the rhombus . In any triangle , the inscribed rhombus has side length , which is independent of .

By construction, and .

Since and is a diagonal of the rhombus , is the angle bisector of at of length .