## Euclid, Elements of Geometry, Book I, Proposition 21 (Edited by Dionysius Lardner, 1855)

Proposition XXI. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]

 (100) The sum of two right lines (D B and D C) drawn to a point (D) within a triangle (B A C) from the extremities of any side (B C), is less than the sum of the other two sides of the triangle (A B and A C), but the lines contain a greater angle.

Produce B D to E. The sum of the sides B A and A E of the triangle B A E is greater than the third side B E (XX); add E C to each, and the sum of the sides B A and A C is greater than the sum of B E and E C, but the sum of the sides D E and E C of the triangle D E C is greater than the third side D C (XX); add B D to each, and the sum of B E and E C is greater than the sum of B D and D C, but the sum of B A and A C is greater than that of B E and E C; therefore the sum of B A and A C is greater than that of B D and D C.

Because the external angle B D C is greater than the internal D E C (XVI), and for the same reason D E C is greater than A, the angle B D C is greater than the angle A.

By the thirty-second proposition it will follow, that the angle B D C exceeds the angle A by the sum of the angles A B D and A C D. For the angle B D C is equal to the sum of D E C and D C E; and, again, the angle D E C is equal to the sum of the angles A and A B E. Therefore the angle B D C is equal to the sum of A, and the angles A B D and A C D.

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