Page:Euclid's Elements 1714 Barrow translation.djvu/31

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EUCLIDE's Elements.

19

PROP. XVIII

The greateſt ſide AC of every triangle ABC ſubtends the greateſt angle ABC.

From AC a take away ADa 3. 1.
b 5. 1.=AB, and join BD. b Therefore is the angle ADB = ABD. But ADB c C; therefore is ABD C; c 16.1.
d 9.ax. d therefore the whole angle ABC C. After the ſame manner, ſhall be ABC A. Which was to he dem.

PROP. XIX.

In every triangle ABC, under the greateſt angle A is ſubtended the greateſt ſide BC.

For if AB be ſuppoſed equal to BC, then will be the angle A a 5.1.a = C, which is contrary to the Hypotheſis : and if AB BC, then ſhall be the angle C bb 18.1. A, which is againſt the Hypotheſis. Wherefore rather BC AB; and after the ſame manner BC AC. Which was to be dem.

PROP. XX.

Of every triangle ABC two ſides BA y AC, any way taken, are greater than the ſide that remains BC.

Produce the line BA, a and a 3. 1.take AD = AC, and draw the line DC, b then ſhall the angle D be equal to b 5. 1.
c 9.ax.
d 19.1.
3 conſtr.
& 2.ax.ACD, c therefore is the whole angle BCDD; d therefore BD (e BA + AC) BC. Which was to be demonſtrated.

PROP. XXI.

If from the utmoſt points of one ſide BC of a triangle ABC two right lines BD, CD be drawn to any point within the triangle, then are both thoſe two lines ſhorier than the two other ſides of

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