## Euclid, Elements of Geometry, Book I, Proposition 48 (Edited by Dionysius Lardner, 1855)

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

 (216) If the square on one side (A C) of a triangle (A B C) be equal to the sum of the squares of the other two sides (A B and B C), the angle (A B C) opposite to that side is a right angle.

From the point B draw B D perpendicular (XI) to one of the sides A B, and equal to the other B C (III), and join A D.

The square of A D is equal to the squares of A B and B D (XLVII), or two the squares of A B and B C which is equal to B D (const.); but the squares of A B and B C are together equal to the square of A C (hyp.), therefore the squares of A D and A C are equal, and therefore the lines themselves are equal; but also D B and B C are equal, and the side A B is common to both triangles, therefore the triangles A B C and A B D are mutually equilateral, and therefore also mutually equiangular, and therefore the angle A B C is equal to the angle A B D; but A B D is a right angle, therefore A B C is also a right angle.

This proposition may be extended thus:

The vertical angle of a triangle is less than, equal to, or greater than a right angle, according as the square of the base is less than, equal to, or greater than the sum of the squares of the sides.

For from B draw B D perpendicular to A B and equal to B C, and join A D.

The square of A D is equal to the squares of A B and B D or B C. The line A C is less than, equal to, or greater than A D, according as the square of the line A C is less than, equal to, or greater than the squares of the sides A B and B C. But the angle B is less than, equal to, greater than a right angle, according as the side A C is less than, equal to, or greater than A D (XXV, VIII); therefore &c.

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