The Midpoint of the Hypotenuse is the Circumcentre of the Right Triangle

The midpoint of the hypotenuse of a right triangle is the circum-centre of the triangle.

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

Let $A\left( {a,0} \right)$, $B\left( {b,0} \right)$ and $C\left( {b,c} \right)$ are any three point on the given circle.
For the point $A\left( {a,0} \right)$, since the point $A$ is on the circle then the equation of circle becomes

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

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

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

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

Centre of the circle is given by

Now midpoint of the hypogenous is given as

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