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 3 of 5: Recognize the perpendicular bisectors

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XY\perp A_1C_1,\quad XZ\perp A_1B_1

Detailed analysis

Because $Q=X$ is the circumcentre of $A_1B_1C_1$ , the lines through $X$ and the arc-midpoint points are exactly the perpendicular bisectors of the corresponding sides. Thus $XY\perp A_1C_1$ and $XZ\perp A_1B_1$ .