# The Midpoint of the Hypotenuse is the Circumcenter of the Right Triangle

The midpoint of the hypotenuse of a right triangle is the circumcenter of the triangle.

Consider the equation of the circle in general form is given by

Let $A\left( {a,0} \right)$, $B\left( {b,0} \right)$ and $C\left( {b,c} \right)$ be any three points on the given circle.

For the point $A\left( {a,0} \right)$, since the point $A$ is on the circle then the equation of the circle becomes

For the point $B\left( {b,0} \right)$, since the point $B$ is on the circle then the equation of the circle becomes

For the point $C\left( {b,c} \right)$, since the point $C$ is on the circle then the equation of the circle becomes

Now solving equation (iv) and equation (iii), we get the value

Solving equation (ii) and equation (iii), we get the value

The center of the circle is given by

Now the midpoint of the hypotenuse is given as

Thus, the midpoint of the hypotenuse is equal to the center of the circle.