## 26. Fortress Wall

A small town, which has the shape of a semicircle, is surrounded by a fortress wall. There are three gates through which one can enter the town; gates $P_{1}$ and $P_{2}$ are located at the ends of the curved section of the wall, while gate $P_{3}$ is located somewhere else on the same curved section of the wall, which has the shape of a semicircle. Roads leading from each gate are directed along the tangents to the semicircle; these roads, obviously, intersect at two points. Another road runs along the straight section of the wall.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-058.jpg?height=542&width=1317&top_left_y=1682&top_left_x=375)

A cyclist rides around the town, traveling along the four indicated roads. He discovered that the square of the duration of the journey $P_{1} P_{2}$ is equal to the product of the duration of the journey $P_{1} P_{3}$ and the duration of the journey $P_{3} P_{2}$. How would you explain this?
26. Let $O$ be the center of the semicircle, and let $A$ and $B$ be the points of intersection of the tangents at points $P_{1}$ and $P_{2}$ with the tangent at point $P_{3}$. Note that $\angle P_{1} A P_{3}$ and $\angle P_{2} B P_{3}$ are obviously supplementary:

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-166.jpg?height=540&width=870&top_left_y=1346&top_left_x=593)

Since $\angle A P_{1} P_{2}$ and $\angle B P_{2} P_{1}$ are obviously right angles, $\angle P_{1} A B + \angle P_{2} B A = 180^{\circ}$. Therefore, $\angle O A P_{1} + \angle O B P_{2} = 90^{\circ}$, since these angles are half of the previously considered angles. Hence, triangles $O A P_{1}$ and $O B P_{2}$ are similar, and consequently,

$$
A P_{1} : O P_{1} = O P_{2} : B P_{2} \text{, or } A P_{1} \cdot B P_{2} = O P_{1} \cdot O P_{2} = O P_{1}^{2}
$$

and

$$
\left(P_{1} A + A P_{3}\right) \cdot \left(P_{3} B + B P_{2}\right) = 4 O P_{1}^{2} = P_{1} P_{2}^{2}
$$

- which is essentially what we needed to prove.