с А, P D T Cor. 1. (a) Let CD become a diameter and be I to AB. Then AP. PB becomes AP, (96°, Cor. 5) B .. AP2=CP.PD, and denoting AP by C, CP by v, and the radius of the circle by r, this becomes C=v(2r-v), which is a relation between a chord of a O, the radius of the O, and the distance CP, commonly called the versed sine, of the arc AB. (6) When the point of intersection P passes without the O we have still, by the principle of con- PA. PB=PC.PD. i.e., if a tangent and a secant be drawn from the same point to a circle, the square on the tangent is equal to the rectangle on the segments of the secant between the point and the circle. Cor. 3. Conversely, if T is on the circle and PT2=PC.PD, PT is a tangent and T is the point of contact. For, if the line PT is not a tangent it must cut the circle in some second point T' (94°). Then PT.PT'=PC.PD=PT2. Therefore PT=PT', which is not true unless T and T' coincide. Hence PT is a tangent and T is the point of contact. Cor. 4. Let one of the secants become a centre-line as PEF. Denote PT by t, PE by h, and the radius of the circle by r. Then PT2=PE.PF becomes =h(2r+h). B F EXERCISES. 1. The shortest segment from a point to a circle is a portion of the centre-line through the point. 2. The longest segment from a point to a circle is a portion of the centre-line through the point. 3. If two chords of a circle are perpendicular to one another the sum of the squares on the segments between the point of intersection and the circle is equal to the square on the diameter. 4. The span of a circular arch is 120 feet and it rises 15 feet in the middle. With what radius is it con structed? 5. A conical glass is b inches deep and a inches across the mouth. A sphere of radius r is dropped into it. How far is the centre of the sphere from the bottom of the glass? 6. The earth's diameter being assumed at 7,960 miles, how far over its surface can a person see from the top of a mountain 3 miles high? 7. How much does the surface of still water fall away from the level in one mile ? 8. Two circles whose radii are 10 and 6 have their centres 12 feet apart. Find the length of their common chord, and also that of their common tangent. 9. Two parallel chords of a circle are c and G1 and their distance apart is d, to find the radius of the circle. 10. If v is the versed sine of an arc, k the chord of half the arc, and r the radius, k2=2vr. 177o. Theorem.-If upon each of two intersecting lines a pair of points be taken such that the rectangle on the segments between the points of intersection and the assumed points in one of the lines is equal to the corresponding rect M D angle for the other line, the four assumed points are concyclic. B (Converse of 176o.) L and M intersect in O, and OA. OB=OC.OD. Then A, B, C, and D are concyclic. Proof.—Since the os are equal, if A and B lie upon the same side of O, C and D must lie upon the same side of 0; and if A and B lie upon opposite sides of O, C and D must lie upon opposite sides of O. Let a o pass through A, B, C, and let it cut M in a second point E. Then OA. OB=OC.OE. (176°) But OA. OB=OC.OD. (hyp.) OD=OE, and as D and E are upon the same side of O they must coincide; .. A, B, C, D are concyclic. q.e.d. T 178°. Let two circles excluding each other without contact have their centres at A and B, and let C be the point, on their common centre-line, which divides AB so that the difference between the squares on the segments AC and CB is equal to the difference between the squares on the con terminous radii. Through C Join AT and BT'. AC2-BC2=AT2- BT2. But, since PC is an altitude in the ΔΑΡΒ, , AC2 - BC2=AP2 – BP, (172°, 1) and AP2=AT2+ PT?, and BP2= BT+PT', (169', Cor. 1) whence PT=PT', and PT=PT. Therefore PCD is the locus of a point from which equal tangents are drawn to the two circles. Def.- This locus is called the radical axis of the circles, and is a line of great importance in studying the relations of two or more circles. Cor. 1. The radical axis of two circles bisects their common tangents. Cor. 2. When two circles intersect, their radical axis is their common chord. Cor. 3. When two circles touch externally, the common tangent at the point of contact bisects the other common tangents. T' 179°. The following examples give theorems of some importance. Ex. I. P is any point without a circle and TT' is the chord of contact (114°, Def.) for the point P. TT' cuts the centre-line PO in Q. Then, PTO being a 7, (110o) OQ. OP=OT. (169) ::. the radius is a geometric mean between the join of any point with the centre and the perpendicular from the centre upon the chord of contact of the point. Def.-P and Q are called inverse points with respect to the circle. Ex. 2. Let PQ be a common direct tangent to the circles having O and Oas centres. Let OP and O'Q be radii to the points of contact, and let QR be || to 00'. Denote A the radii by r and r'. Then AC=00'+r-r', BD=00'-ptri. .:. AC.BD=002- (r-)=QR2-PRP=PQ? (169", Cor. 1) P R D B Similarly it may be shown that EXERCISES. 1. The greater of two chords in a circle is nearer the centre than the other. 2. Of two chords unequally distant from the centre the one nearer the centre is the greater. 3. AB is the diameter of a circle, and P, Q any two points on the curve. AP and BQ intersect in C, and AQ and BP in C'. Then AP.AC+BQ.BC=AC'. AQ+BC'.BP. 4. Two chords of a circle, AB and CD, intersect in O and are perpendicular to one another. If R denotes the radius of the circle and E its centre, 8R2=AB2+CD2 + 40E?. 5. Circles are described on the four sides of a quadrangle as diameters. The common chord of any two adjacent circles is parallel to the common chord of the other two. 6. A circle S and a line L, without one another, are touched by a variable circle 2. The chord of contact of Z passes through that point of S which is farthest distant from L. 7. ABC is an equilateral triangle and P is any point on its circumcircle. Then PA+PB+PC=o, if we consider the line crossing the triangle as being negative. 8. CD is a chord parallel to the diameter AB, and P is any point in that diameter. Then PC2+ PDP=PA2+ PB?. |