$$\eqalign{\mbox{Area of Square}&= (a+b)^2 \cr &=
    a^2+2ab+b^2}$$<br />
    $$\eqalign{\mbox{Area of Trapezium}&amp;= \frac{1}{2}\times
    \mbox{Area of Square} \cr &amp;=\frac{1}{2}\times(a^2+2ab+b^2)
    \cr &amp;= \frac{a^2}{2} + ab + \frac{b^2}{2}}$$<br />
    I can also work out the area of the trapezium using the three
    right angled triangles:<br />
    $$\eqalign{\mbox{Area of three right angled triangles} &amp;=
    \frac{ab}{2}+\frac{ab}{2}+\frac{c^2}{2} \cr &amp;= ab +
    \frac{c^2}{2}} $$<br />
    I can equate the two expressions for the area of the trapezium
    and simplify:<br />
    $$\eqalign{\frac{a^2}{2} + ab + \frac{b^2}{2}&amp;=ab +
    \frac{c^2}{2}\cr \Rightarrow \frac{a^2}{2}+\frac{b^2}{2} &amp;=
    \frac{c^2}{2}}$$<br />
    Then multiplying both sides of the equation by $2$ gives
    $$a^2+b^2=c^2$$<br />
    <br />