2. In what case would the construction fail, if the equilateral triangle were described on the other side of DE? Exercises. 1. Prove this Proposition without using Prop. viii. 2. Prove that AF is perpendicular to DE. 3. Prove that any point in AF is equally distant from the points D and E. 4. Prove that any point in AF is equally distant from the lines AB, AC. PROP. X.—Problem. To bisect a given finite right line (AB). Sol.—Upon AB describe an equilateral triangle ACB [i.]. Bisect the angle ACB by the line CD [ix.], meeting AB in D, then AB is bisected in D. Dem.—The two triangles ACD, BCD, have the side AC equal to BC, being the sides of an equilateral triangle, and CD common. Therefore the two sides AC, CD in one are equal to the two sides BC, CD in the other; and the angle ACD is equal to the angle BCD (const.). Therefore the base AD is equal to the base DB [iv.]. Hence AB is bisected in D. Exercises. 1. Show how to bisect a finite right line by describing two circles. 2. Every point equally distant from the points A, B is in the line CD. PROP. XI.—Problem. From a given point (C) in a given right line (AB) to draw a right line perpendicular to the given line. Sol.—In AC take any point D, and make CE equal to CD [iii.]. Upon DE describe an equilateral triangle DFE [i.]. Join CF. Then CF shall be at right angles to AB. Dem.—The two triangles DCF, ECF have CD equal to CE (const.) and CF common; therefore the two sides CD, CF in one are respectively equal to the two sides CE, CF in the other, and the base DF is equal to the base EF, being the sides of an equilateral triangle (Def. xxi.); therefore [viii.] the angle DCE is equal to the angle ECF, and they are adjacent angles. Therefore (Def. xiii.) each of them is a right angle, and CF is perpendicular to AB at the point C. Exercises. 1. The diagonals of a lozenge bisect each other perpendicularly. 2. Prove Prop. xi. without using Prop. viii. 3. Erect a line at right angles to a given line at one of its extremities without producing the line. 4. Find a point in a given line that shall be equally distant from two given points. 5. Find a point in a given line such that, if it be joined to two given points on opposite sides of the line, the angle formed by the joining lines shall be bisected by the given line. 6. Find a point that shall be equidistant from three given points. PROP. XII.—Problem. To draw a perpendicular to a given indefinite right line (AB) from a given point (C) without it. Sol.—Take any point D on the other side of AB, and describe (Post. iii.) a circle, with C as centre, and CD as radius, meeting AB in the points F and G. Bisect FG in H [x.]. Join CH (Post. i.). CH shall be at right angles to AB.