Euclid, Elements of Geometry, Book I, Proposition 10
(Edited by Dionysius Lardner, 1855)

Proposition X. Problem.
[Euclid, ed. Lardner, 1855, on Google Books]

(71) To bisect a given right line (A B).


Upon the given line A B describe an equilateral triangle A C B (I), A B C D bisect the angle A C B by the right line C D (IX); this line bisects the given line in the point D.


Because the sides A C and C B are equal (const.), and C D common to the triangles A C D and B C D, and the angles A C D and B C D also equal (const.); therefore (IV) the bases A D and D B are equal, and the right line A B is bisected in the point D.

In this and the following proposition an isosceles triangle would answer the purposes of the solution equally with an equilateral. In fact, in the demonstrations the triangle is contemplated merely as isosceles: for nothing is inferred from the equality of the base with the sides.

Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)

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