Let ∆ABC be the right-angled triangle in which \[\angle B\text{ }=\text{ }{{90}^{o}}\]
Let \[\mathbf{AC}\text{ }=\text{ }\mathbf{x},\text{ }\mathbf{BC}\text{ }=\text{ }\mathbf{y}\]
In this way, \[\mathbf{AB}\text{ }=\text{ }\surd \left( \mathbf{x2}\text{ }+\text{ }\mathbf{y2} \right)\]
\[\angle \mathbf{ACB}\text{ }=\text{ }\mathbf{\theta }\]
\[\mathbf{Let}\text{ }\mathbf{z}\text{ }=\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{y}\text{ }\left( \mathbf{given} \right)\]
Presently, the space of \[\mathbf{ABC}\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\mathbf{x}\text{ }\mathbf{AB}\text{ }\mathbf{x}\text{ }\mathbf{BC}\]
Thusly, the space of the given triangle is greatest when the point between its hypotenuse and a side is π/3.