Consider a right triangle with integer sides having no common factor. Show that the average of the odd leg and the hypotenuse is a square number.
The odd leg of a right triangle is of the form $\displaystyle m^2-n^2 $ and in your case $\displaystyle (m,n)=1 $. The hypotenuse of the same triangle is then $\displaystyle m^2+n^2 $.
Now what is $\displaystyle \frac{(m^2+n^2)+(m^2-n^2)}{2} $?