Taking @Frank000's suggestion to give a full answer for those that wish to see one possible method.
Using the fact that $$\displaystyle \frac{c}{2} - \frac{ab}{c} = 7$$ we rearrange to find
$$
c^2 -14c = 2ab.
$$
Note also that $a+b=72-c$, squaring this gives $$ a^2 + 2ab + b^2 = 72^2 -144c +c^2,$$
since $a^2 + b^2 =c^2$ and we have an expression for $2ab$,
$$
c^2 + c^2 - 14c = 72^2 -144c +c^2,
$$
or rather
$$
c^2 + 130c - 72^2 = 0,
$$
the positive solution of which is $c=32$, giving $ a = \displaystyle \frac{288}{b}$, and substitution into the pythagorean theorem gives $$\frac{288^2}{b^2} + b^2 = 32^2,$$ which gives $a = 4(5 \pm 7)$, $b = 4(5 \mp 7)$.
This problem could probably also be done with multiple applications of Heron's formula, summing the area of the some of the interior triangles.