Euclid, Elements of Geometry, Book I, Proposition 18
(Edited by Sir Thomas L. Heath, 1908)

Proposition 18
[Euclid, ed. Heath, 1908, on]

In any triangle the greater side subtends the greater angle.

For let ABC be a triangle having the side AC greater than AB;
I say that the angle ABC is also greater than the angle BCA.

For, since AC is greater than AB, let AD be made equal to AB [I. 3] , and let BD be joined.


Then, since the angle ADB is an exterior angle of the triangle BCD,
it is greater than the interior and opposite angle DCB. [I. 16]

But the angle ADB is equal to the angle ABD,
since the side AB is equal to AD;
therefore the angle ABD is also greater than the angle ACB;
therefore the angle ABC is much greater than the angle ACB.

Therefore etc. Q.E.D.

Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)

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