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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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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Cushion Ball Interactivity

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

This interactivity is designed for the problem Cushion Ball.

The interactive diagram below has two labelled points, A and B. What is the shortest path from A to B if you bounce off one cushion? In the diagram, you can click on the "Show" buttons to draw the four possible paths from A to B. Which is the shortest? You may move A and B around by clicking on them.

What is the shortest path from A to B using exactly two cushions? The interactive diagram below shows the eight possible paths from A to B. How would you calculate the shortest path?