Can you make sense of the three methods to work out the area of the kite in the square?

Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?