Hypotenuse

The hypotenuse of a right triangle is the side opposite the right angle. It is also the longest side of the triangle.

By the Pythagorean theorem, the length of the hypotenuse of a triangle with legs of length $a$ and $b$ is $\sqrt{a^2 + b^2}$.

For any right triangle, the hypotenuse is a diameter of the circumcircle. It follows that the midpoint of the hypotenuse of the triangle is the center of the circle. The converse also holds: if the length of the median of $\triangle ABC$ from $C$ is the same as $\frac12 AB$, then $\triangle ABC$ is a right triangle with its right angle at $C$.

See also

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