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 2 of 5: Identify the circumcentre with X

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Q=X

Detailed analysis

Let $Q$ be the circumcentre of $A_1B_1C_1$ . The hypothesis puts $Q$ on $\Omega$ . By the cyclic symmetry of the construction, we may choose the labeling so that $\angle B_1A_1C_1>90^\circ$ . Then $A$ and the arc-midpoint point $X$ lie on the same side of $B_1C_1$ . Since $Q$ lies on the perpendicular bisector of $B_1C_1$ and also on $\Omega$ , the configuration forces $Q=X$ .