Draw the diagonal DB; then in the triangles ABD, BCD, we have AB=DC, AD=BC, and BD com mon; hence these triangles are equal (Prop. 22), and the angles ABD, ADB, of the one, respectively equal to the angles BDC, DBC, of the A Fig. 27. D B other; and, consequently, AB is parallel to DC, and AD to BC (Prop. 10), and the figure is a parallelogram. PROP. XXVI. THEOREM. If two of the opposite sides of a quadrilateral are equal and parallel, the other two sides will be equal and parallel, and the figure will be a parallelogram. In the quadrilateral ABCD (preceding diagram), let AB be equal and parallel to DC; then we have to prove that AD is equal and parallel to BC. The line DB cutting the parallels AB, DC, makes the alternate angles ABD, BDC, equal (Prop. 9); hence the triangles ABD, CDB, having the two sides AB, BD, and the included angle ABD of the one, equal respectively to the sides CD, BD, and the included angle BDC of the other, are equal (Prop. 4), AD=BC, the angles ADB, CBD, equal, and consequently AD parallel to BC (Prop. 10); hence the figure is a parallel ogram. PROP. XXVII. THEOREM. The diagonals of a parallelogram mutually bisect each other. We have to prove that AI-IC, and DI=IB. The angles ACD, CAB, are equal, and the angles BOOK II. DEFINITIONS. 1. EVERY line which is not straight, is called a curve line. 2. A circle is a space enclosed by a curve line, every point of which is equally distant from a point within ; which point is called the centre. 3. The boundary of a circle is called its circumfer ence. 4. A radius is a line drawn from the centre to the circumference. All radii are equal. 5. A diameter is a line which passes through the centre, and has its extremities in the circumference. ameter therefore is double the radius. In the circle enclosed by the circumference AEFBD, C is the centre, CD is a radius, and AB is a diameter. 6. An arc is any portion of the circumference, as EFG. A F A di G 7. The chord of an arc is a straight line joining its extremities. EG is the chord of the arc EFG. 8. A segment of a circle is the portion included between an arc and its chord. The space EFG, between the arc EFG and the chord EG, is a seg ment. 9. A sector of a circle is the portion included between two radii. BDC, the part of the circle between BC, CD, and the arc BD, is a sector. 10. A secant is a line cutting the circumference of the circle, and lying partly within and partly without the circle. 11. A tangent is a line which touches the circumference in only one point, which is called the point of contact. AB is a secant, and DE a tangent to the circle whose centre is C. 12. One circle touches another when their circumferences have only one point in common. 13. A line is inscribed in a circle when its extremities are in the circumference; as the line AB. 14. An angle is inscribed, when its vertex is in the circumference; as the angle A. C B 15. A triangle is inscribed, when the vertices of its three angles are in the circumference; as the triangle ABC. 16. Generally any polygon is inscribed, when the vertices of all its angles are in the circumference; the circle is then said to circumscribe the polygon. 17. A circle is inscribed in a polygon, when its circumference touches each side of the polygon; and the polygon is then said to circumscribe the circle. 18. Equal circles are those which are described with equal radii. PROP. I. THEOREM. Every diameter divides the circle and its circumference into two equal parts. To prove this, suppose the portion ADB applied upon the semi-circle AEB; the curve line ADB must fall the circle and its circumference are divided into equal parts by the diameter AB. Cor. Hence in equal circles, the semi-circles and semi-circumferences are respectively equal. PROP. II. THEOREM. Every chord is less than the diameter. We have to prove that the chord AD is less than the diameter AB. Draw CD. Now in the triangle ADC, AD<AC+ CD (Book I. Prop. 16); but CD=CB (Def. 4); hence AD AC+CB; that is, AD AB. Fig. 30. A B |