Let ABC be an acute scalene triangle with no angle equal to 60∘. Let ω be the circumcircle of ABC. Let ΔB be the equilateral triangle with three vertices on ω, one of which is B. Let ℓB be the line through the two vertices of ΔB other than B. Let ΔC and ℓC be defined analogously. Let Y be the intersection of AC and ℓB, and let Z be the intersection of AB and ℓC.
Let N be the midpoint of minor arc BC on ω. Let R be the triangle formed by ℓB, ℓC, and the tangent to ω through N. Prove that the circumcircle of AYZ and the incircle of R are tangent.
