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

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The greatet ide AC of every triangle ABC ubtends the greatet angle ABC.

From AC a take away AD=AB, and join BD. b Therefore is the angle ADB = ABD. But ADB c C; therefore is ABD  C;  d therefore the whole angle ABC  C. After the ame manner, hall be ABC  A. Which was to he dem.

In every triangle ABC, under the greatet angle A is ubtended the greatet ide BC. For if AB be uppoed equal to BC, then will be the angle A a = C, which is contrary to the Hypotheis : and if AB BC, then hall be the angle C b A, which is againt the Hypotheis. Wherefore rather BC AB; and after the ame manner BC  AC. Which was to be dem.

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

Produce the line BA, a and take AD = AC, and draw the line DC, b then hall the angle D be equal to ACD, c therefore is the whole angle BCDD; d therefore BD (e BA + AC) BC. Which was to be demontrated.

If from the utmot points of one ide BC of a triangle ABC two right lines BD, CD be drawn to any point within the triangle, then are both thoe two lines horier than the two other ides of