2019 AIME I Problem 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.