Let ABC be an acute scalene triangle with no angle equal to 60°. Let ω be the circumcircle. Let l_B and l_C be lines formed by vertices of equilateral triangles inscribed in ω with one vertex at B and C respectively. Let Y = AC ∩ l_B, Z = AB ∩ l_C. Let R be the triangle formed by l_B, l_C, and the tangent at the midpoint of minor arc BC. Prove the circumcircle of AYZ and the incircle of R are tangent.

Step 1 (Geometric Setup): Let $\omega$ be the circumcircle of acute scalene triangle $ABC$. The lines $l_B$ and $l_C$ are chord lines of equilateral triangles inscribed in $\omega$, passing through $B$ and $C$ respectively. The line $t$ is the tangent to $\omega$ at the midpoint of minor arc $BC$. The region $R$ is the triangle bounded by $l_B$, $l_C$, and $t$. Let $\gamma$ be the incircle of $R$.

Step 2 (Miquel Point and Radical Center): Let $Y=AC\cap l_B$ and $Z=AB\cap l_C$. By properties of Miquel's Theorem, the circumcircle of $AYZ$ passes through the Miquel point $T$ of the configuration. Angle chasing around the equilateral triangles shows that the angles formed by $l_B$ and $l_C$ are symmetric w.r.t. the bisector of $\angle A$.

Step 3 (Homothetic Tangency): We set up a homothety centered at $A$ that maps the incircle $\gamma$ of $R$ to the circumcircle of $AYZ$. By showing that the center of homothety preserves the distance ratio of the tangency point to the respective lines, we establish that the circumcircle of $AYZ$ and the incircle of $R$ are tangent. $■$