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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.

1 2 3 4 5

Proposition 6.

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;

1 (meaning the equal sides). Cf. note on the similar expression in Prop. 4, lines 2, 3.

2 , εἰλήφθω ἐπὶ τῆς ΒΔ τυχὸν σημεῖον τὸ Ζ, where τυχὸν σημεῖονmeans “a chance point.”

3 δύο αἱ ΖΑ, ΑΓ δυσὶ ταῖς ΗΑ, ΑΒ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ. Here, and in numberless later passages, I have inserted the word “sides” for the reason given in the note on I. 1, line 20. It would have been permissible to supply either “straight lines” or “sides” ; but on the whole “sides” seems to be more in accordance with the phraseology of I. 4.

4 i.e., apparently, common to the angles, as the αὐτῶν in βάσις αὐτῶν κοινὴ can only refer to γωνία and γωνίᾳ preceding. Simson wrote “and the base BC is common to the two triangles BFC, CGB” ; Todhunter left out these words as being of no use and tending to perplex a beginner. But Euclid evidently chose to quote the conclusion of I. 4 exactly; the first phrase of that conclusion is that the bases (of the two triangles) are equal, and, as the equal bases are here the same base, Euclid naturally substitutes the word “common” for “equal.”

5 As “(Being) what it was required to prove” (or “do” ) is somewhat long, I shall henceforth write the time-honoured “Q. E. D.” and “Q. E. F.” for ὅπερ ἔδει δεῖξαι and ὅπερ ἔδει ποιῆσαι.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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