The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

PROPOSITION III. THEOREM.

THat Line which divideth the Angle of a Triangle, into two equal parts, divideth its Base in two parts, which are in the same Reason to each other as are their Sides. And if that Line divideth the Base into parts proportional to the Sides, it shall divide the Angle into Two equally.

If the Line AD divideth the Angle BAC into Two equal parts; there shall be the same Reason of AB to AC, as of BD to DC. Continue the Side CA, and make AE equal to AB; then draw the Line EB.

Demonstration. The exterior Angle CAB is equal to the Two interior Angles AEB, ABE; which being equal (by the 5th. of the 1st.) seeing the Sides AE, AB, are equal; the Angle BAD, the half of BAC, shall be equal to one of them; that is to say to the Angle ABE. Thence (by the 27th. of the 1st.) the Lines AD, EB, are parallel; and (by the 2d.) there is the same Reason of EA, or AB to AC, as of BD to DC.

Secondly. If there be the same Reason of AB to AC, as of BD to DC, the Angle BAC shall be divided into Two equally.

Demonstra. There is the same reason of AB or AE to AC, as of BD to DC: thence the Lines EB, AD, are parallel; and (by the 29th. of the 1st.) the Alternate Angles EBA, BAD, the internal BEA, and the external DAC, shall be equal; and the Angles EBA, AEB, being equal; the Angles BAD, DAC, shall be so likewise. Wherefore the Angle BAC hath been divided equally.

USE.

WE make use of this Proposition to attain to the Proportion of the sides.