Kitap 4
3
But the angles DEG, DEF are also equal to two right angles; [I. 13] therefore the angles AKB, AMB are equal to the angles DEG, DEF, of which the angle AKB is equal to the angle DEG; therefore the angle AMB which remains is equal to the angle DEF which remains.
3
Similarly it can be proved that the angle LNB is also equal to the angle DFE; therefore the remaining angle MLN is equal to the angle EDF. [I. 32]
3
Therefore the triangle LMN is equiangular with the triangle DEF; and it has been circumscribed about the circle ABC.
3
Therefore about a given circle there has been circumscribed a triangle equiangular with the given triangle. Q. E. F. at random, literally as it may chance, ὡς ἕτυχεν. The same expression is used in III. 1 and commonly. is in fact divisible, καὶ διαιρεῖται, literally is actually divided.
PROPOSITION 4.
4
In a given triangle to inscribe a circle.
4
Let ABC be the given triangle; thus it is required to inscribe a circle in the triangle ABC.
4
Let the angles ABC, ACB be bisected by the straight lines BD, CD [I. 9], and let these meet one another at the point D; from D let DE, DF, DG be drawn perpendicular to the straight lines AB, BC, CA.
