If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.
For in the circle ABCD let the two straight lines AC, BD cut one another at
the point E; I say that the rectangle contained by AE, EC is equal to the
rectangle contained by DE, EB.
If now AC, BD are through the centre, so that E is the centre of the circle
ABCD, it is manifest that, AE, EC, DE, EB being equal,
the rectangle contained by AE, EC is also equal to the rectangle contained by
DE, EB.
Next let AC, DB not be through the centre; let the centre of ABCD be taken,
and let it be F;
from F let FG, FH be drawn perpendicular to the straight lines AC, DB,
and let FB, FC, FE be joined.
Then, since a straight line GF through the centre cuts a straight line AC not
through the centre at right angles, it also bisects it;
therefore AG is equal to GC.
Since, then, the straight line AC has been cut into equal parts at G and into
unequal parts at E, the rectangle contained by AE, EC together with the square
on EG is equal to the square on GC;
Let the square on GF be added;
Therefore the rectangle AE, EC together with the squares on GE, GF
is equal to the squares on CG, GF.
But the square on FE is equal to the squares on EG, GF,
and the square on FC is equal to the squares on CG, GF;
therefore the rectangle AE, EC together with the square on FE
is equal to the square on FC.
And FC is equal to FB;
therefore the rectangle AE, EC together with the square on EF
is equal to the square on FB.
For the same reason, also, the rectangle DE, EB together with the square on FE
is equal to the square on FB.
But the rectangle AE, EC together with the square on FE was also proved equal
to the square on FB; therefore the rectangle AE, EC together with the square on
FE
is equal to the rectangle DE, EB together with the square on FE.
Let the square on FE be subtracted from each; therefore the rectangle
contained by AE, EC which remains is equal to the rectangle contained by DE, EB.
Therefore, etc;
Q.E.D.