11b. Construct a Triangle Given the Lengths of Two Sides and the Bisector of Their Common Angle

This Demonstration shows an alternative construction of a triangle given the lengths of the sides and and the length of the angle bisector of .

Construction

Draw the line segment with interior point so that and .

Step 1: Draw parallel lines through and at an arbitrary angle . On the line through , measure out so that . Let be the intersection of and the parallel line through .

Step 2: Construct an isosceles triangle with base and a leg of length .

Step 3: Let be a point such that and . That is, is a parallelogram. Let point be the intersection of the lines and .

Verification

In the triangle , divides in the ratio . So is the bisector of the angle at .