I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was done and was wondering if someone could point me towards this? I believe it was first done by Gage?
The final paper: Grayson, Matthew A.(1UCSD) Shortening embedded curves. Ann. of Math. (2) 129 (1989), no. 1, 71–111.
using results from:
Gage, Michael E.(1CWR) An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50 (1983), no. 4, 1225–1229.
Gage, M. E.(1CWR) Curve shortening makes convex curves circular. Invent. Math. 76 (1984), no. 2, 357–364.
Gage, M.(1RCT); Hamilton, R. S.(1UCSD) The heat equation shrinking convex plane curves. J. Differential Geom. 23 (1986), no. 1, 69–96.
Grayson, Matthew A.(1UCSD) The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 (1987), no. 2, 285–314.

$\begingroup$ Hmm. The original post asked for the isoperimetric inequality in the euclidean plane. So, while Grayson's Annals paper is very nice, I don't think it is needed here. The JDG paper is of course necessary to deal with the nonconvex case. $\endgroup$ – Sam Nead Jul 9 at 9:12
There is a sketch of the argument you want in the first paragraph of Section 5 of "The heat equation shrinking convex plane curves" by Gage and Hamilton. But the general result requires curve shortening for nonconvex curves, and that is due to Grayson. However Grayson does not say this. So the "proof" is spread over four journal articles.