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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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For What?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Cosines Rule

Age 14 to 16 Challenge Level:

(a) Prove that a triangle with sides $3$, $7$, $8$ contains a $60$ degree angle.

(b) Three points $A$, $B$ and $C$ lie in this order on a line, and $P$ is any point in the plane. Use the Cosine Rule to prove that:

\[{AP^2\over AB.AC}+{CP^2\over CA.CB} = 1 + {PB^2\over BA.BC}.\]