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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

Why do this problem ?

In this one problem you meet many important aspects of mathematics. It illustrates how much mathematics is inter-related. It shows the value of not being content to find one solution, but of asking yourself "could I solve this another way?" or "have I found the best method?" It is very satisfying to feel you have somehow got to the very essence of a mathematical idea by looking at it in the right way.

Also this problem provides a good example of how you can generalise from a result that is really a simple case of a much more general result. Are you content just to solve a problem or do you ask yourself "what if ...." and try to find more general results?

Possible approach

You could challenge your class to find as many different methods of solving this problem asposible. You could tell them that one pair of school students found 8 different methods. At some stage of this work you might mention that these two students used respectively sines, cosines, tangents, vectors, matrices, coordinate geometry, complex numbers and pure geometry.

Perhaps your students could work in pairs. They could come to the board and present their solutions to the rest of the class and/or make posters for the classroom wall. You could see collectively how many different methods your class can find.

Key questions

What lengths in the diagram can we find?

What do we know about sines, cosines, tangents, vectors, matrices, coordinate geometry, complex numbers and pure geometry that we can use to prove this result?

How might we generalise this result?

Possible extension

The linked article Why Stop at Three by One? beautifully generalises this result.