PROPOSITION 8. THEOREM.
If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square on the other part, is equal to the square on the straight line which is made up of the whole and that part.
Let the straight line AB be divided into any two parts at the point C: four times the rectangle AB, BC, together with the square on AC shall be equal to the square on the straight line made up of AB and BC together.
Produce AB to D, so that BD may be equal to CB; [Post. 2. and I. 3.
on AD describe the square AEFD;
and construct two figures such as in the preceding propositions.
Then, because CB is equal to BD,[Construction.
and that CB is equal to GK, and BD to KN,[I. 34.
therefore GK is equal to KN.[Axiom 1.
For the same reason PR is equal to RO.
And because CB is equal to BD, and GK to KN, the rectangle CK is equal to the rectangle BN, and the rectangle GR to the rectangle RN.[I. 36.
But CK is equal to RN, because they are the complements of the parallelogram CO;[I. 43.
therefore also BN is equal to GR.[Axiom 1.
Therefore the four rectangles BN, CK, GR, RN are equal to one another, and so the four are quadruple of one of them CK.
Again, because CB is equal to BD, [Construction.
and that BD is equal to BK, [II. 4, Corollary.
that is to CG, [I. 34.
and that CB is equal to GK, [I. 34.
