Question 11
In $\triangle ABC,$ the sides have integer lengths and $AB = AC.$ Circle $\omega$ has its center at the incenter of $\triangle ABC.$ An *excircle* of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to $\overline{BC}$ is internally tangent to $\omega,$ and the other two excircles are both externally tangent to $\omega.$ Find the minimum possible value of the perimeter of $\triangle ABC.$