· A tangent is often considered as terminating at one end at the point of contact, at the other where it meets another tangent or a secant. 10. A Secant (in geometry) is a line lying partly within and partly without a circle; as G E. A secant is generally considered as terminating at one end where it meets the concave circumference, and at the other where it meets another secant or a tangent. THEOREM I. 11. In the same circle, or equal circles, equal angles at the centre are subtended by equal arcs; and, conversely, equal arcs subtend equal angles at the centre. the angle E; as they are equal they will coincide; and as BA and BC are equal to ED and E F, the point A will coincide with D, and the point C with F; and the arc A C will coincide with DF, otherwise there would be points in the one or the other arc unequally distant from the centre. Conversely. If the arcs AC and D F are equal, the angles B and E are equal. For, if the radius AB is placed on the radius D E with the point B on E, the point A will fall on D, as A B = DE; and the arc AC will coincide with D F, otherwise there would be points in the one or the other arc unequally distant from the centre; and as the arc A C D F, the point C will fall on F; therefore BC will coincide with EF, and the angle B be equal to E. THEOREM II. 12. In the same or equal circles, equal chords subtend equal arcs; and, conversely, equal arcs are subtended by equal chords. equal; and conversely, if the chords A B and DE are equal, the arcs A B and D E are equal. For, if the centre of the circle A B C is placed on the centre of D E F with the point A of the circumference on the point D, if the arcs or the chords are equal, B will fall on E; and in either case the chords and arcs will coincide, otherwise there would be points in the one or the other circumference unequally distant from the centre. THEOREM III. 13. Angles at the centre vary as their corresponding arcs. Let ACD, DCE, ECF be equal angles at the centre C; then the arcs A D, DE, E F are equal (11); then the angle ACE, being double the angle ACD, the arc AE is double the arc AD; and the angle AC F, being three times the angle AC D, the arc A F is three times H G F E D A E 224 the arc AD; and the angle ACG, being m times the angle AC D, the arc AG is m times the arc AD; that is, the angle varies as the arc, or the arc as the angle. 14. Cor. 1. As angles at the centre vary as their arcs, or arcs as their corresponding angles, either of these quantities may be assumed as the measure of the other. The measure of an angle is, then, the arc included between its sides and described from its vertex as a centre. 15. Cor. 2. As the sum of all the angles about the point C is equal to four right angles (I. 9), one right angle, HC A, is measured by one quarter of the circumference, or by a quadrant. THEOREM IV. 16. The radius perpendicular to a chord bisects the chord and the arc subtended by the chord. Let CE be a radius perpendicular to the chord AB; it bisects the chord A B, and also the arc A E B. Draw the radii CA and CB and the chords AE and EB. As equal oblique A lines are equally distant from the perpen C B E dicular, A D = DB (I. 52); and as E is a point in the perpendicular to the middle of A B, it is equally distant from A and B (I. 53); therefore the chords and the arcs AE, E B are equal. 17. Corollary. The perpendicular to the middle of a chord passes through the centre of the circle. DEFINITIONS. 18. An Inscribed Angle is one whose vertex is in the circumference and whose sides are chords; as DA B. 19. An Inscribed Polygon is one whose sides are chords. Thus ABCDEF is inscribed in the outer circle. 20. A Circumscribed Polygon is one whose sides are tangents. Thus ABCDEF is circumscribed about the inner circle. In this case the circle is said to be inscribed in the polygon. B In this case C O E F D THEOREM V. 21. An inscribed angle is measured by half the arc included by its sides. 1st. When one of the sides BD is a diameter; then the angle B is measured by half the arc A D. Draw the radius CA, and the triangle ACB is, isosceles, CA and C B being radii; therefore the angle AB (I. 42). But the exterior angle ACD is equal to the sum of the two angles A and B (I. 39); therefore the B D angle ACD is equal to half the angle B; the angle 4 CD is measured by the arc A D (14); therefore the angle B is measured by half the arc A D. 2d. When the centre is within the angle, draw the diameter B C. By the preceding part of the proposition the angle ABC is measured by half the arc AC, and CBD by half CD; therefore ABC CBD, or ABD, is measured. by half ACCD, or the arc A D. A B D 24. Every equilateral polygon inscribed in a circle is regular. Let ABCDEF be an equilateral polygon inscribed in a circle; it is also equiangular and therefore regular. For the chords AB, BC, CD, &c. being equal, the arcs AB, BC, CD, &c. are equal (12); therefore the arc AB the arc BC will be equal to the arc BC B C E D the arc CD, &c.; that is, the angles B, C, &c. are in equal segments; therefore they are equal (22), and the polygon is equiangular and regular. THEOREM VII. 25. An infinitely small chord coincides with its arc. Let A B be an infinitely small chord; it coincides with the arc AD B. |