3. In triangle $A B C$, point $D$ is the foot of the altitude on side $A B$. Points $E$ are on segment $A D$ and $F$ is on side $B C$, such that $\angle B A F = \angle A C E$. Segments $A F$ and $C E$ intersect at point $G$, and segments $A F$ and $C D$ intersect at point $T$. Determine the sizes of the angles of triangle $A B C$ if triangle $C G F$ is equilateral and triangle $A E T$ is isosceles with vertex $E$.
III/3. Let $\angle B A F=\angle A C E=\varphi$. Since triangle $C G F$ is equilateral, it follows that $\angle C G A=120^{\circ}$. Therefore, $\angle G A C=180^{\circ}-\angle C G A-\angle A C G=60^{\circ}-\varphi$ and we have $\angle B A C=\angle B A F+\angle G A C=\varphi+60^{\circ}-\varphi=$ $60^{\circ}$.

Since triangle $A E T$ is isosceles, $\angle E T A=\varphi=$ $\angle E C A$, so points $A, E, T$ and $C$ are concyclic. Then $\angle E C T=\angle E A T=\varphi$. Therefore, $2 \varphi=\angle A C E+$ $\angle E C D=\angle A C D=\frac{\pi}{2}-\angle D A C=30^{\circ}$, from which it follows that $\varphi=15^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_2dd3bccc19bc96d6d7dfg-13.jpg?height=517&width=631&top_left_y=1963&top_left_x=1204)

We can also calculate $\angle C B A=\angle F B A=60^{\circ}-\varphi=45^{\circ}$ and $\angle A C B=75^{\circ}$.
Calculation of $\angle F A C=60^{\circ}-\varphi$ ..... 1 point
Conclusion $\angle B A C=60^{\circ}$ ..... 1 point
Points $A, E, T$ and $C$ are concyclic ..... 1 point
Conclusion $\angle E C T=\varphi$ ..... 1 point
Calculation $\varphi=15^{\circ}$ ..... 1 point
Conclusion $\angle C B A=45^{\circ}$ ..... 1 point
Conclusion $\angle A C B=75^{\circ}$ ..... 1 point