English: Construction by New Orleans students Calcea Johnson and Ne’Kiya Jackson that yields a trigonometric proof of the Pythagorean theorem for a<b. ^[1]

The proof is valid for almost all right triangles, except for the case when a=b, i.e. Isosceles Right Triangle.

For the case when ${\displaystyle a\neq b}$, we can always choose to orient the right triangle ${\displaystyle \triangle ABC}$ so that a<b as shown in the figure. Then we reflect the right triangle to obtain point D. We extend leg BE perpendicular to AB, and extend AD until it crosses BE, here is why it is necessary for ${\displaystyle 2\alpha <{\frac {\pi }{2}}}$. Then, we need to sum a convergent infinite geometric series where ${\displaystyle {\frac {a^{2}}{b^{2}}}<1}$ in order to compute ${\displaystyle \sin \left(2\alpha \right)}$ from the large right triangle ${\displaystyle \triangle ABE}$:

${\displaystyle \sin \left(2\alpha \right)={\frac {\overline {BE}}{\overline {AE}}}={\frac {2c{\frac {a}{b}}\left(1+{\frac {a^{2}}{b^{2}}}+\left({\frac {a^{2}}{b^{2}}}\right)^{2}+\left({\frac {a^{2}}{b^{2}}}\right)^{3}+\ldots \right)}{c+2c{\frac {a^{2}}{b^{2}}}\left(1+{\frac {a^{2}}{b^{2}}}+\left({\frac {a^{2}}{b^{2}}}\right)^{2}+\left({\frac {a^{2}}{b^{2}}}\right)^{3}+\ldots \right)}}={\frac {2c{\frac {a}{b}}\left({\frac {1}{1-{\frac {a^{2}}{b^{2}}}}}\right)}{c+2c{\frac {a^{2}}{b^{2}}}\left({\frac {1}{1-{\frac {a^{2}}{b^{2}}}}}\right)}}={\frac {2ab}{a^{2}+b^{2}}}}$

Finally, we employ the Law of sines in ${\displaystyle \triangle ABD}$ to find out:

${\displaystyle {\frac {2a}{\sin \left(2\alpha \right)}}={\frac {c}{\sin \beta }}}$

which upon substituion with ${\displaystyle \sin \beta ={\frac {b}{c}}}$ and ${\displaystyle \sin \left(2\alpha \right)={\frac {\overline {BE}}{\overline {AE}}}}$ gives the Pythagorean theorem:

${\displaystyle a^{2}+b^{2}=c^{2}}$.

For the special case when ${\displaystyle a=b}$, the geometric series does not converge because ${\displaystyle {\frac {a^{2}}{b^{2}}}=1}$, however, the proof is purely algebraic using the areas of triangles ${\displaystyle \triangle ABC}$, ${\displaystyle \triangle ACD}$ and ${\displaystyle \triangle ABD}$, namely:

${\displaystyle {\frac {ab}{2}}+{\frac {ab}{2}}={\frac {c^{2}}{2}}}$, but since ${\displaystyle a=b}$, it follows that ${\displaystyle {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}={\frac {c^{2}}{2}}}$ and hence ${\displaystyle a^{2}+b^{2}=c^{2}}$.