(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

Let ABC be a triangle
having the angle ABC equal to
the angle ACB;

I say that the side AB is
also equal to the side AC.

For, if AB is unequal to AC, one of them is greater.

Let AB be greater; and
from AB the greater
let DB
be cut off equal to AC
the less;

let DC be joined.

Then, since DB is equal
to AC, and
BC is common,

the two sides DB,
BC are equal to the
two sides AC,
CB respectively;

and the angle DBC is equal
to the angle ACB;

therefore the base DC is equal
to the base AB,

and the triangle DBC will be equal
to the triangle ACB,

the less to the greater;

which is absurd.

Therefore AB is not unequal
to AC;

it is therefore equal to it.

Therefore etc. Q.E.D.

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

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