HMMT February

Let $\alpha, \beta, \gamma$ denote the measures of $\frac{1}{2} \angle A, \frac{1}{2} \angle B, \frac{1}{2} \angle C$, respectively. We have $m \angle C E F=90^{\circ}-\gamma, m \angle F E A=90^{\circ}+\gamma, m \angle A F G=m \angle A F E=180^{\circ}-\alpha-\left(90^{\circ}+\gamma\right)=$ $\beta=m \angle A B G$, so $A B F G$ is cyclic. Now $A G=G F$ implies that $\overline{B G}$ bisects $\angle A B F$. Since $\overline{B G}$ by definition bisects $\angle A B C$, we see that $F$ must lie on $\overline{B C}$. Hence, $F=D$. If $I$ denotes the incenter of triangle $A B C$, then $\overline{I D}$ is perpendicular to $\overline{B C}$, but since $A, I, F$ are collinear, we have that $\overline{A D} \perp \overline{B C}$. Hence, $A B C$ is isoceles with $A B=A C$. Furthermore, $B C=2 B F=2$. Moreover, since $A B F G$ is cyclic, $\angle B G A$ is a right angle. Construct $F^{\prime}$ on minor $\operatorname{arc} G F$ such that $B F^{\prime}=6$ and $F^{\prime} G=1$, and let $A B=x$. By the Pythagorean theorem, $A F^{\prime}=B G=\sqrt{x^{2}-36}$, so that Ptolemy applied to $A B F^{\prime} G$ yields $x^{2}-36=x+36$. We have $(x-9)(x+8)=0$. Since $x$ is a length we find $x=9$. Now we have $A B=A C=9$. Pythagoras applied to triangle $A B D$ now yields $A D=\sqrt{9^{2}-1^{2}}=4 \sqrt{5}$, which enables us to compute $[A B C]=\frac{1}{2} \cdot 2 \cdot 4 \sqrt{5}=4 \sqrt{5}$. Since the area of a triangle is also equal to its semiperimeter times its inradius, we have $4 \sqrt{5}=10 r$ or $r=\frac{2 \sqrt{5}}{5}$.