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 1 of 5: Use the arc-midpoint lemma

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BC_1=CB_1,\quad XC_1=XB_1,\quad X=\text{midpoint of the major arc }BC

Detailed analysis

Let the circle $(AB_1C_1)$ meet $\Omega=(ABC)$ again at $X$ . The excircle tangent-length relations give $BC_1=CB_1$ . Therefore the spiral similarity at $X$ taking $\triangle XBC_1$ to $\triangle XCB_1$ is a congruence, so $XC_1=XB_1$ . Hence $X$ is the midpoint of the major arc $BC$ of $\Omega$ . Define $Y$ and $Z$ analogously for the other two vertices; $Y$ and $Z$ lie on the perpendicular bisectors of $A_1C_1$ and $A_1B_1$ , respectively.