Right Triangle

A Triangle with an Angle of 90° ( radians). The sides , , and of such a Triangle satisfy the Pythagorean Theorem. The largest side is conventionally denoted and is called the Hypotenuse.

For any three similar shapes on the sides of a right triangle,

 (1)

which is equivalent to the Pythagorean Theorem. For a right triangle with sides , , and Hypotenuse , let be the Inradius. Then
 (2)

Solving for gives
 (3)

But any Pythagorean Triple can be written
 (4) (5) (6)

so (5) becomes
 (7)

which is an Integer when and are integers.

Given a right triangle , draw the Altitude from the Right Angle . Then the triangles and are similar.

In a right triangle, the Midpoint of the Hypotenuse is equidistant from the three Vertices (Dunham 1990). This can be proved as follows. Given , let be the Midpoint of (so that ). Draw , then since is similar to , it follows that . Since both and are right triangles and the corresponding legs are equal, the Hypotenuses are also equal, so we have and the theorem is proved.

See also Acute Triangle, Archimedes' Midpoint Theorem, Brocard Midpoint, Circle-Point Midpoint Theorem, Fermat's Right Triangle Theorem, Isosceles Triangle, Malfatti's Right Triangle Problem, Obtuse Triangle, Pythagorean Triple, Quadrilateral, RAT-Free Set, Triangle

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.

Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120-121, 1990.