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) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) |

so (5) becomes

(7) |

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.

**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.

© 1996-9

1999-05-25