## 34. Alsatian Village

The main street of the village (it is called Main) is perpendicular to General de Gaulle Street; they intersect at the Town Hall Square. The Catholic cathedral faces the main street, while the Protestant one faces General de Gaulle Street. The school is located on the street connecting these two cathedrals, at a bird's flight distance (500 m) from each of the mentioned streets. All streets are straight. One day, the teacher asked his students[^9] to calculate the sum of the reciprocals of the distances (in meters) from the town hall to each of the cathedrals. The students did not know trigonometry or analytic geometry.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-061.jpg?height=654&width=1213&top_left_y=455&top_left_x=433)

What did they answer their teacher?
34. Let $E$ be the school, $M$ the town hall, $P$ the Protestant church, and $C$ the Catholic church. Since the area of triangle $M C P$ is equal to the sum of the areas of triangles $M C E$ and $M P E$,

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-171.jpg?height=808&width=1177&top_left_y=1361&top_left_x=445)

$500 M C / 2 + 500 M P / 2 = M P \cdot M C / 2$,

i.e.,

$$
M C + M P = M P \cdot M C / 500
$$

## Therefore,

$$
1 / M C + 1 / M P = 1 / 500 = 0.002
$$

170