If on the circumference of a circle two points be taken at random
, the straight line joining the points will fall within the circle
be a circle, and let two points A
be taken at random on its circumference; I say that the straight line joined from A
will fall within the circle.
For suppose it does not, but, if possible, let it fall outside, as AEB
; let the centre of the circle ABC
be taken [III. 1
], and let it be D
; let DA
be joined, and let DFE
be drawn through.
Then, since DA
is equal to DB
, the angle DAE is also equal to the angle DBE. [I. 5]
And, since one side AEB
of the triangle DAE
is produced, the angle DEB is greater than the angle DAE. [I. 16]
But the angle DAE
is equal to the angle DBE
; therefore the angle DEB is greater than the angle DBE.
And the greater angle is subtended by the greater side; [I. 19
] therefore DB is greater than DE. But DB is equal to DF;
therefore DF is greater than DE,
the less than the greater : which is impossible.
Therefore the straight line joined from A
will not fall outside the circle.
Similarly we can prove that neither will it fall on the circumference itself; therefore it will fall within.
Therefore etc. Q. E. D.