Prove that a circle described on the hypotenuse of a right angle triangle as diameter , passes through the opposite - Maths - Circles

Question

Prove that a circle described on the hypotenuse of a right angle
triangle as diameter , passes through the opposite angular point

Priyanka Kedia · Accepted Answer

Let ABC is a right angled triangle with $∠ B = 90^{0}$ .

To Prove : Circle drawn with hypotenuse as diametre passes through the opposite vertex B.

Proof : Let us assume that the circle drawn with AC as diametre does not pass through B. but meets AB produced at point D.

Join DC.

$∠ A D C = 90^{0} (a n g l e i n s e m i c i r c l e) a n d, ∠ A B C = 90^{0} (g i v e n) \Rightarrow ∠ A D C = ∠ A B C = 90^{0} T h i s i s i m p o s s i b l e u n l e s s t h e p o int s B a n d D c o i n c i d e$

Thus, our assumption that the circle drawn with AC as diametre does not pass through point B is false.

Hence, Circle drawn with hypotenuse as diametre passes through the opposite vertex B

Prove that a circle described on the hypotenuse of a right angle triangle as diameter , passes through the opposite angular point.