Coroll.
Hence, every equilateral triangle is alſo equiangular.
PROP. VI.
An image should appear at this position in the text. A high-res raw scan of the page is available. To use it as-is, as a placeholder, edit this page and replace "{{missing image}}" with "{{raw image|Euclid's Elements 1714 Barrow translation.djvu/24}}". If it needs to be edited first (e.g. cropped or rotated), you can do so by clicking on the image and following the guidance provided. [Show image]  | |
If two angles ABC, ACB of a triangle ABC be equal the one to the other, the ſides AC, AB ſubtended under the equal angles, ſhall alſo be equal one to the other.
If the ſides be not equal, let one be bigger than the other, ſuppoſe BA 
CA. a a 3. 1.
b 1. poſt. Make BD = CA, and b draw the line CD.
In the triangles DEC, ACB, becauſe BD cc ſuppoſ.
d hyp.
e 4. 1.
f 9. ax. = CA, and the ſide BC is common, and the angle DBC d = ACB, the triangles DBC, ACB e ſhall be equal the one to the other, a part to the whole. f Which is impoſſible.
Coroll.
Hence, Every equiangular triangle is alſo equilateral.
PROP. VII.
An image should appear at this position in the text. A high-res raw scan of the page is available. To use it as-is, as a placeholder, edit this page and replace "{{missing image}}" with "{{raw image|Euclid's Elements 1714 Barrow translation.djvu/24}}". If it needs to be edited first (e.g. cropped or rotated), you can do so by clicking on the image and following the guidance provided. [Show image] | |
Upon the ſame 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 ſame points A, B, on the ſame ſide C, to ſeveral points, as C and D, but only to C.
1. Caſe. If the point D be ſet in the line AC, it is plain that AD is aa 9. ax. not equal to AC.
2. Caſe. 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.
