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{{ printedBook.courseTrack.name }} {{ printedBook.name }} If a right triangle is inscribed in a circle, then the hypotenuse is a diameter. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

The Inscribed Right Triangle Theorem is also know as the **Thales' Theorem**.

Consider a right triangle $ABC,$ with $m∠B=90_{∘},$ inscribed in a circle.

According to the Inscribed Angle Theorem, the measure of $∠B$ is half the measure of its intercepted arc, $AC.$

Substituting $m∠B=90_{∘}$ into the equation above gives that $mAC=180_{∘}.$ Then, $AC$ is a semicircle, implying that $AC$ $($the hypotenuse of $△ABC)$ is a diameter of the circle.

For the converse, consider a triangle $ABC$ inscribed in a circle such that one side of the triangle is a diameter of the circle.

Since $AC$ is a diameter, then $AC$ is a semicircle and then $mAC=180_{∘}.$ Now, the *Inscribed Angle Theorem* gives a relation between this arc and the angle opposite to the diameter.
$m∠B =21 mAC=21 (180_{∘})=90_{∘} $
The last equation implies that $∠B$ is a right angle which makes $△ABC$ a right triangle.