Proof of Isoperimetric Inequality using Curve Shortening Flow 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?
 A: 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 non-convex curves, and that is due to Grayson.  However Grayson does not say this.  So the "proof" is spread over four journal articles. 
A: The final paper:
Grayson, Matthew A.(1-UCSD)
Shortening embedded curves.
Ann. of Math. (2) 129 (1989), no. 1, 71–111. 
using results from:
Gage, Michael E.(1-CWR)
An isoperimetric inequality with applications to curve shortening.
Duke Math. J. 50 (1983), no. 4, 1225–1229. 
Gage, M. E.(1-CWR)
Curve shortening makes convex curves circular.
Invent. Math. 76 (1984), no. 2, 357–364. 
Gage, M.(1-RCT); Hamilton, R. S.(1-UCSD)
The heat equation shrinking convex plane curves.
J. Differential Geom. 23 (1986), no. 1, 69–96. 
Grayson, Matthew A.(1-UCSD)
The heat equation shrinks embedded plane curves to round points.
J. Differential Geom. 26 (1987), no. 2, 285–314. 
