The sum of distances from the sides of a regular polygon to an interior point is a constant

Question

Let there be a regular polygon of $n$ sides. Assume there is a point $P$ inside the polygon, then prove that

$$a_1 + a_2 + a_3 + \cdots + a_n= \text{constant}$$ where $a_i$ is the distance of the point from the $i$-th side.

Please help as soon as possible.....

Answer 1

The area of the triangle formed by the point and the $i$-th side is proportional to $a_i$. These areas add up to the area of the polygon, so the $a_i$ also add up to a constant (the area over half the side length).