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

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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.

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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 .

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 .

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Contributed by: Izidor Hafner (October 2017)
Open content licensed under CC BY-NC-SA


Snapshots


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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