the supposition is false; .. that AC does 2. that similarly AC does not fall on the O, Part 2nd. That between AE and ○ no right line can be drawn which does not cut . FE H B D Suppose that FA is between them without cutting the O; and prove, on that supposition, 1. that, in ▲ DAG, DA > DG, 2, that DH > DG, i. e., less > greater, which shows the supposition to be false. PROPOSITION XVII. Problem. To drawn a straight line from a given point, either G without or in the circumference, which shall touch a given circle. Part 1st. To draw a tangent to O BCD, from point A without it. 2. 3. that EBA is a right, that AB touches the O BCD. Part 2nd, in which given point D is on the of the O, is proved by Cor. to xvi. 3. Steps of the Demonstration. Suppose that Fc is not line drawn from F, as FG, is that supposition, DE. DE, but that some other 1. that GCF <rt. ▲ FGC, 2. that FC > FG, 3. that FB > FG, i. e., less > greater; which shows the supposition to be false, and . FG is not DE, 4. that, similarly, no line but Fc, drawn from F, is DE. but Suppose that the centre of the O is not in AC, in some other direction, as F; and prove, on that supposition, B 1. that FCE is a rt. ▲, 2. that FCE = ACE, i. e., less = greater, which shows that the supposition is false, and .. F is not the centre, 3. that the centre is in AC. PROPOSITION XX. Theorem. The angle at the centre of a circle is double of the angle at the circumference, upon the same base, that is, upon the same part of the circumference. Steps of the Demonstration to Case 1st, In which the centre E is within BAC. that 2 EAB = ex. ▲ BEF of ▲ BAE, 3. 4. that similarly 2 ≤ EAC = FEC, that :. whole ▲ BEC = 2 whole ▲ BAC. B Steps of the Demonstration to Case 2nd, E B 1. Prove that 2. that 2 EDC GEC, 3. 4. that similarly 2 ≤ gdb = ≤ geb, that remaining ▲ BEC = 2 remaining E BDC. PROPOSITION XXI. Theorem. The angles in the same segment of a circle are equal to one another. Steps of the Demonstration to Case 1st, 1. Prove that BFD2 BAD, 2. 3. that similarly that. BAD bfd = 2 ≤ bed, BED. E |