24a. Construct a Triangle Given the Length of the Altitude to the Base, the Difference of Base Angles and the Sum of the Lengths of the Other Sides

This Demonstration shows a marked-ruler (or verging) construction of a triangle given the length of the altitude to the base, the difference of angles at the base and the sum of the lengths of the other two sides.

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 .