MathLabs

Problem 3

Let the excircle of triangle ABCABC opposite the vertex AA be tangent to the side BCBC at the point A1A_1. Define the points B1B_1 on CACA and C1C_1 on ABAB analogously, using the excircles opposite BB and CC, respectively. Suppose that the circumcentre of triangle A1B1C1A_1B_1C_1 lies on the circumcircle of triangle ABCABC. Prove that triangle ABCABC is right-angled.
Step 4 of 5: Chase the decisive angle
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∠A=∠C1XB1=∠C1XA1+∠A1XB1=2∠YXA1+2∠A1XZ=2∠YXZ=180∘−∠A\angle A=\angle C_1XB_1=\angle C_1XA_1+\angle A_1XB_1=2\angle YXA_1+2\angle A_1XZ=2\angle YXZ=180^\circ-\angle A
Detailed analysis

The arc-midpoint relations and the two perpendicular bisectors give the directed-angle chain ∠A=∠C1XB1=∠C1XA1+∠A1XB1=2∠YXA1+2∠A1XZ=2∠YXZ=180∘−∠A\angle A=\angle C_1XB_1=\angle C_1XA_1+\angle A_1XB_1=2\angle YXA_1+2\angle A_1XZ=2\angle YXZ=180^\circ-\angle A.