Problem 3

Let the excircle of triangle

ABC

opposite the vertex

A

be tangent to the side

BC

at the point

A_1

. Define the points

B_1

CA

and

C_1

AB

analogously, using the excircles opposite

B

and

C

, respectively. Suppose that the circumcentre of triangle

A_1B_1C_1

lies on the circumcircle of triangle

ABC

. Prove that triangle

ABC

is right-angled.

Step 5 of 5: Conclude that ABC is right-angled

2\angle A=180^\circ\implies\angle A=90^\circ

Detailed analysis

The final equality gives $2\angle A=180^\circ$ , hence $\angle A=90^\circ$ . Therefore $ABC$ is right-angled.