THEOREM 6. A tangent meets the circle in one point only, viz. the point of contact. For since a secant can cut a circle in two points only, it follows that the parts AB, CD are wholly without the. circle; and therefore when C moves up to B, and the chord BC is merged in the point B, the whole line, with exception of the point B, is outside the circle. THEOREM 7. The radius to the point of contact is at right angles to the tangent. For if F be the middle point of the chord BC, OF is perpendicular to BC; and as C moves to B, F will also move up to B, and when the secant becomes a tangent, OF, which is always at right angles to the secant, coincides with the radius OB. Therefore OB is at right angles to the tangent TBS. COR. I. a given point. COR. 2. The line at right angles to the tangent through the point of contact passes through the centre. Def. 16. When a secant is drawn from the point of contact of a tangent it divides the circle into segments which are said to be alternate to the angles made by the tangent with the secant on its sides opposite to the segments. W. G. H Thus ACB is a segment alternate with BAS, and BDA with BAT. THEOREM 8. If from the point of contact of a straight line and a circle a chord of the circle be drawn, the angles made by the chord with the tangent will be equal to the angles in the alternate segments of the circle. Let a chord AB be drawn from the point of contact A of the tangent TAS. Then will the angles BAS, BAT be equal to the angles in the alternate segments of the circle. For take any point P in T the arc of the segment alternate to BAS, and join PB, PA, and produce PA to Q. R -S Conceive the point P to move along the arc towards A. Then the angle BPA in the segment remains always the same; and it will after a while assume the position of the dotted lines BP'A, and ultimately when P has moved up to A, the angle BPA will coincide with the angle BAS, since the limiting position of the secant PQ is the tangent AS, and BP then coincides with BA. Therefore the angle BAS= the angle BPA in the alternate segment. Similarly it may be shewn that the angle BAT= the angle BRA. Def. 17. If from any point on a curve a line is drawn at right angles to the tangent at that point, it is called a normal to the curve at that point. Since the radius is at right angles to the tangent to a circle, it follows that all radii are normals to a circle. THEOREM 9. From any point within or without a circle except the centre, two and only two normals can be drawn, one of which is the shortest, and the other the longest line that can be drawn from that point to the circumference: and as a point moves along the circumference from the extremity of the shortest to the extremity of the longest normal, its distance from the fixed point continually increases. B C B Let A be the fixed point, O the centre, and let AO produced through the centre meet the circumference in B, and produced if necessary in the other direction meet it in C. Then AB and AC are normals and are the only normals, and are respectively the longest and shortest lines. that can be drawn from A to the circumference: and if P, Q are any other points such that the arc CP is less than CQ, AP shall be less than AQ. Join OP, OQ. Then it is clear that AB and AC are at right angles to the tangents at B and C, since A is a point in the radius OB or OC. And if P is any other point, OP is the normal at that point, and therefore AP is not the normal: hence two and only two normals can be drawn. Again, in the triangle APO, the difference of OP and OA is AC, since OP = OC; and therefore AC is less than AP (1. 13): and the sum of OP and OA is AB, since OP=OB, and therefore AB is greater than AP. Hence of all lines drawn from A to the circumference AC is the least and AB the greatest. Lastly, in the triangles AOP, AOQ since the two sides AO, OP are equal to AO, OQ, but the angle AOQ greater than the angle AOP, therefore AQ is greater than AP (I. 17). Therefore as a point moves from C to A along the arc, its distance from A continually increases. COR. I. Two and only two equal straight lines can be drawn from A to the circumference, one on each side of the shortest normal. COR. 2. A point from which more than two equal straight lines can be drawn to a circumference must be the centre. THEOREM 10. INTERSECTION OF CIRCLES. The line that joins the centres of two intersecting circles, or that line produced, bisects at right angles their common chord. Let O, O' be the centres of two intersecting circles, PQ their common chord, then shall 00′ or OƠ′ produced bisect PQ at right angles. For since PQ is a chord of both circles, the line which bisects PQ at right angles passes through both centres (II. 2, ẞ); that is, it must be the line OO. CONTACT OF CIRCLES. Def. 18. When one of the points in which one circle cuts another moves up to and ultimately coincides with the other, the circles are said to touch one another at that point. Since two circles intersect in only two points, it follows that two circles which touch one another can have no other point in common; for the two points of intersection are merged in the point of contact. THEOREM II. If two circles touch one another, the line that joins their centres will pass through the point of contact. For if in the figures of Th. 10, the centres O, O' of the circles were to recede from one another, or were to approach one another, the points P and Q would after a while approach one another, and the chord PQ would become indefinitely small, and be merged in the point 7, and the circles would touch one another at T by the definition. |