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Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

Three by One

Stage: 5 Challenge Level: Challenge Level:1

The solutions produced here by school students show eight different methods. Reflection on these methods will help other students to see something of the 'bigger picture' in a way they will not experience from ploughing through the syllabus and working from textbooks (although that is also absolutely necessary).

Eight distinct proofs were given to this problem by two students, Alex and Neil (Madras College) using respectively sines, cosines, tangents, vectors, matrices, coordinate geometry, complex numbers and pure geometry.

See Alex and Neil's solution here.

Alex and Neil went on to generalise this problem to rectangles with dimensions $n$ by 1.

See 'Why Stop at Three by One?'