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?

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?

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.)

Direct Logic

Stage: 5 Challenge Level:

To prove a theorem directly we start with something known to be true and then proceed, making small logical steps which are clearly correct, until we arrive at the desired result. So, because the starting point was true and each small step clearly correct, we know the result to be true.

Breaking down a mathematical argument into small steps requires patience and clear thinking.

In the following interactivities we have written out three proofs, broken them into small steps and then shuffled up the steps. Can you rearrange them into the correct logical order?

Proof of the formula for the roots of a quadratic equation

Proof of the formula for the sum of an arithmetic progression

Proof of the formula for the sum of a geometric progression