[Math Lair] Elements: Book I, Proposition 5

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The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 1. This well-known demonstration is known in Latin as the pons asinorum or "bridge of fools", possibly because the construction looks like a bridge, and because, as this is the first proposition in the Elements to require a fair bit of thought to prove, it may have been used in antiquity to separate those knowledgeable in geometry from those who are not.

Elements on the Math Lair
Book IBook IIBook IX
Definitions, Postulates, and Common NotionsDefinitionsProposition 20
Proposition 1, Proposition 3, Proposition 14Proposition 36
Proposition 5, Proposition 6,  
Proposition 29, Proposition 47  

Proposition 5.

In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

Let ABC be an isosceles triangle having the side AB equal to the side AC;
and let the straight lines BD, CE be produced further in a straight line with AB, AC. [Post. 2]

I say that the angle ABC is equal to the angle ACB, and the angle CBD to the angle BCE. [diagram]

Let a point F be taken at random on BD;
from AE the greater let AG be cut off equal to AF the less; [I. 3]

and let the straight lines FC, GB be joined.[Post. 1]

Then, since AF is equal to AG and AB to AC,
  the two sides FA, AC are equal to the two sides GA, AB, respectively;
and they contain a common angle, the angle FAG.

Therefore the base FC is equal to the base GB,
and the angle AFC to the angle AGB. [I. 4]
And, since the whole AF is equal to the whole AG,
and in these AB is equal to AC,
the remainder BF is equal to the remainder CG.

But FC was also proved equal to GB;
therefore the two sides BF, FC are equal to the two sides CG, GB respectively;
and the angle BFC is equal to the angle CGB,

while the base BC is common to them;
therefore the triangle BFC is also equal to the triangle CGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend;
therefore the angle FBC is equal to the angle GCB,
and the angle BCF to the angle CBG.

Accordingly, since the whole angle ABG was proved equal to the angle ACF,

and in these the angle CBG is equal to the angle BCF,
the remaining angle ABC is equal to the remaining angle ACB;
and they are at the base of the triangle ABC.
But the angle FBC was also proved equal to the angle GCB;
and they are under the base.

Therefore etc. Q.E.D.