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 theHypotenuse.
For any three similar shapes on the sides of a right triangle,
 | (1) |
, , and Hypotenuse, let 
be the Inradius. Then | (2) |
gives | (3) |
 |  |  | (4) |
 | |  | (5) |
 | |  | (6) |
so (5) becomes
 | (7) |
and 
are integers.
Given a right triangle 
, draw the Altitude 
from the Right Angle 
. Then thetriangles 
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 alsoequal, 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.
- Euclidean Plane
- Euclidean Space
- Euclid Number
- Euclid's Axioms
- Euclid's Elements
- Euclid's Fifth Postulate
- Euclid's Postulates
- Euclid's Principle
- Euclid's Theorems
- Eudoxus's Kampyle
- Euler Angles
- Euler-Bernoulli Triangle
- Euler Brick
- Euler Chain
- Euler Characteristic
- Euler Constant
- Euler Curvature Formula
- Euler Differential Equation
- Euler Equation
- Euler Formula
- Euler Four-Square Identity
- Euler Graph
- Eulerian Circuit
- Eulerian Cycle
- Eulerian Integral of the First Kind