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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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Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Proof Sorter - Sum of an AP

Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?
This is a self-checking and enjoyable way to learn and to understand the proof of the formula

Possible approach
Not everyone has their maths lessons in a computer room! This exercise can be set for homework if the learners have access to computers at school to do their homework, or at home.

If you want to use this actvity without a computer how about clicking on 'Printable Page' and printing this for yourself. You can then cut out the pieces and rearrange the bits of paper. Of course teachers can photo-copy this for a class.

Key question
Can you explain all the steps in the proof?