Page:Euclid's Elements 1714 Barrow translation.djvu/24

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Hence, every equilateral triangle is alo equiangular.

If two angles ABC, ACB of a triangle ABC be equal the one to the other, the ides AC, AB ubtended under the equal angles, hall alo be equal one to the other.

If the ides be not equal, let one be bigger than the other, uppoe BA CA. a  Make BD = CA, and b draw the line CD.

In the triangles DEC, ACB, becaue BD c = CA, and the ide BC is common, and the angle DBC d = ACB, the triangles DBC, ACB e hall be equal the one to the other, a part to the whole. f Which is impoible.

Hence, Every equiangular triangle is alo equilateral.

Upon the ame right line AB two right lines being drawn AC, BC, two other right lines equal to the former, AD, BD, each to each (viz. AD = AC, and BD = BC) cannot be drawn from the ame points A, B, on the ame ide C, to everal points, as C and D, but only to C.

1. Cae. If the point D be et in the line AC, it is plain that AD is a not equal to AC. 2. Cae. If the point D be placed within the triangle ACB,then draw the line CD,and produce BDF, and BCE. Now you would have AD = AC.