Book 4
4
Therefore in the given triangle ABC the circle EFG has been inscribed. Q. E. F. and distance one of the (straight lines D)E, (D)F, (D)G. The words and letters here shown in brackets are put in to fill out the rather careless language of the Greek. Here and in several other places in Book IV. Euclid says literally and with distance one of the (points) E, F, G (καὶ διαστήματι ὲνὶ τῶν E, Z, H) and the like. In one case (IV. 13) he actually has with distance one of the points G, H, K, L, M (διαστήματι ὲνὶ τῶν Η, Θ, Κ, Λ, Μ σημείων). Heiberg notes Graecam locutionem satis miram et negligentem, but, in view of its frequent occurrence in good MSS., does not venture to correct it.
PROPOSITION 5.
5
About a given triangle to circumscribe a circle.
5
Let ABC be the given triangle; thus it is required to circumscribe a circle about the given triangle ABC.
5
Let the straight lines AB, AC be bisected at the points D, E [I. 10], and from the points D, E let DF, EF be drawn at right angles to AB, AC; they will then meet within the triangle ABC, or on the straight line BC, or outside BC.
5
First let them meet within at F, and let FB, FC, FA be joined.
5
Then, since AD is equal to DB, and DF is common and at right angles, therefore the base AF is equal to the base FB. [I. 4]
5
Similarly we can prove that CF is also equal to AF; so that FB is also equal to FC; therefore the three straight lines FA, FB, FC are equal to one another.
