Within, an equilateral triangle whose length of each side is 200 meters. Five metropolitan police officers guard the garden taking a position as far away from each other as possible to cover more. Using the Pigeon-Hole Principle, explain how with five police officers, there are always two police officers within 100 meters of each other. You may use a picture to support your proof. You may also assume the following theorems are true.
- The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.
- The distance between two points inside an equilateral triangle is less than the side of the triangle.
What I have tried: depending on the two theorems I have, there are four parts but even if 5 officers falls like this and 2 of them are near, I dont get how it is possible tbat there's only 2 of them that falls near and not 3? Shouldn't the one adjacent to both of them be near too?
Hint: For each pair of sides, join their midpoints to each other by a line segment. This makes four sub-triangles, which are the pigeon holes, and the five police officers are the pigeons. Use the two facts you are given.