Euclid,

ElementsI 21©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 21: If two straight-lines are constructed within a triangle from the limits on one of the sides, the constructed straight-lines will be smaller than the remaining two sides of the triangle and enclose a larger angle.

(diagram 1) Let two straight-lines, BD, DG, be constructed within a triangle, ABG, on one of the sides, BG, fro the limits, B, G. I say that BD, DG are smaller than the remaining two sides of the triangle, BA, AG, and they enclose a larger angle, that by BDG, than that by BAG. (diagram 2) For let BD be drawn through to E. (diagram 3) And since the two sides of any triangle are larger than the remaining, therefore, the two sides, AB, AE, of triangle ABE are larger than BE. (diagram 4) Let a common be added, EG. Therefore, BA, AG are larger than BE, EG. (diagram 5) Again, since the two sides, GE, ED, of triangle GED are larger than GD, (diagram 6) let a common be added, DB. Therefore, GE, EB are larger than GD, DB. (diagram 7) But BA, AG were shown larger than BE, EG. Therefore, BA, AG are much larger than BD, DG. (this clarifies the inequality: diagram 8)

(diagram 9) Again, since the exterior angle of any triangle is larger than the interior and opposite, therefore, the exterior angle, that by BDG, of triangle GDE is larger than that by GED. (diagram 10) Hence, for the same reasons, the exterior angle, that by GEB, of triangle ABE is larger than that by BAG. (diagram 11) But the angle by DBG was shown larger than that by GEB. Therefore, the angle by BDG is much larger than that by BAG. Therefore, if two straight-lines are constructed within a triangle from the limits on one of the sides, the constructed straight-lines will be smaller than the remaining two sides of the triangle and enclose a larger angle, just what it was required to show.

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