Crescents and Triangles

Using Pythagoras` theorem in $ABC$,

$$\begin{eqnarray}a &=& \sqrt{b^2 + c^2} \\ \mbox{Area of semicircle on BC} &=& \pi \frac {a^2}{4}\\ &=& S1\\ \mbox{Area of semicircle on AC} &=& \pi \frac {b^2}{4}\\ &=& S2 \\ \mbox{Area of semicircle on AB} &=& \pi \frac{c^2}{4}\\ &=& S3\\ \mbox{Area of crescents} &=& S2 + S3 + \mbox{Area ABC} - S1\\ &=& \pi \frac {b^2}{4} +\pi \frac {c^2}{4} + \mbox{Area ABC} - \pi \frac {a^2}{4}\\ &=& \pi \frac {b^2}{4} + \pi \frac{c^2}{4} - \pi \frac {a^2}{4} + \mbox{Area ABC}\\ &=& \frac{\pi}{4} \times ( b^2 + c^2 - a^2 ) + \mbox{Area ABC}\\ &=& 0 + \mbox{Area ABC} \\ &=& \mbox{Area ABC} \end{eqnarray}$$