For let FK be made equal to BC, and FL equal to D.
Then, since FK is equal to BC, and of these the part FH is equal to the part BG,
therefore the remainder HK is equal to the remainder GC.
And since, as EF is to D, so is D to BC, and BC to A,
while D is equal to FL, BC to FK, and A to FH,
therefore, as EF is to FL, so is LF to FK, and FK to FH.
Separando, as EL is to LF, so is LK to FK, and KH to FH. [VII. 11, 13]
Therefore also, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents; [VII. 12]
therefore, as KH is to FH, so are EL, LK. KH to LF, FK, HF.
But KH is equal to CG, FH to A, and LF, FK, HF to D, BC, A;
therefore, as CG is to A, so is EH to D, BC, A.
Therefore, as the excess of the second is to the first, so is the excess of the last to all those before it.
Q. E. D.
Complete text
Book I