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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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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?

Tracking Points

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
By viewing the 3-D coordinates as if you could not see the third dimension you can extend what is happening in the 2D case and this is explored in the problem Coordinate Patterns first published in September 2004.