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 .
Triangles and are congruent, so , and .

THINGS TO TRY

SNAPSHOTS

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DETAILS

This is a verging, or marked-ruler construction. For verging, see [1, p. 124].
Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.
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