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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?

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

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Kite in a Square

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