# Double Application of Apollonius' Theorem

- Author:
- Paul Miller

Just as the square of the RADIUS (half the DIAGONAL of the parallelogram) plus the square representing half of ITS SECOND DIAGONAL form a fixed sum equal to half the sums of squares on the SIDES of the parallelogram, so the squares of the SIDES of the half parallelogram (a similar figure) sum to half of the squares of RADIUS plus SECOND DIAGONAL
The 4 : 1 PRODUCT of these ratios is obvious from similarity of the smaller parallelogram to the larger parallelogram and is as a result of applying Apollonius' Theorem successively.