2019 Putnam A2

In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle.

Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively.

Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2 \tan^{-1} (1/3)$. Find $\alpha$.

在三角形$\triangle ABC$中，设$G$为质心，$I$为内接圆的圆心。

令 $\alpha$ 和 $\beta$ 分别为顶点 $A$ 和 $B$ 处的角度。

假设线段 $IG$ 与 $AB$ 平行，且 $\beta = 2 \tan^{-1} (1/3)$。找到$\alpha$。