Problem 7.5.1. Points $A, B, C, D, E, F$ on the diagram satisfy the following conditions:

- points $A, C, F$ lie on the same line;
- $A B=D E=F C$
- $\angle A B C=\angle D E C=\angle F C E$
- $\angle B A C=\angle E D C=\angle C F E$
- $A F=21, C E=13$.

Find the length of segment $B D$.

![](https://cdn.mathpix.com/cropped/2024_05_06_ce96a31ee4d4d7b89169g-09.jpg?height=509&width=734&top_left_y=850&top_left_x=366)
# Answer: 5.

Solution. Note that triangles $A B C, D E C$ and $F C E$ are equal by the second criterion of triangle congruence.

Since $\angle D E C=\angle F C E$, lines $D E$ and $A F$ are parallel. Therefore, $\angle A C B=\angle C D E=$ $\angle C A B$, so all three triangles are isosceles, and their lateral sides are equal to 13.

It remains to calculate the answer:

$$
B D=B C-C D=13-A C=13-(A F-C F)=13-(21-13)=5 .
$$