Page:The New International Encyclopædia 1st ed. v. 04.djvu/862

* CIRCLE. 758 CIRCLE. radical axis is the line at infinity; therefore, a sj'steni of concentric circles passes tlirovijih two imaginary points at infinity. These are called the circular points. The radical axis of two circles is the locus of points from -which tangents to the two circles are equal. If a variable polygon inscribed in a circle of a co-axal system moves so that all the sides but one touch fixed circles of the system, the last side also touches, in every position, a fixed circle of the system ( Poncelet's theorem ). Fio. 4. (2) Inversion. — Let O (Fig. 4) be the centre of a circle of radius r, and P, Q two jioints on a line through O, such that OPOQ = r. P and Q are called inverse points with respect to the circle. Either point is said to be the inverse of the other. The circle and its centre are called the circle and centre of inversion, and )■ the constant of inversion. If every point of a plane figure be inverted with res|K'et to a circle, or every point of a figure in respect to a sphere, the resulting figure is called the inverse image of the given one. The inverse of a circle is either a straight line or a circle, according as the centre of inversion is or is not on the given circle. The centre of inversion is then the centre of similitude of the original circle and its in- verse; and the circle, its inverse, and the circle of inversion are coaxal. The theory of inversion was invented by Stubbs and Ingram in 1842, and has l)eeii made use of by Lord Kelvin in several important propositions of nuithematical physics. (3) Pole and Polar. — Tlic polar of any point P, with respect to a circle, is the peri)endicular to the diameter OP drawn through the inverse point. ITcnce the polar of a point exterior to a circle is the chord joining the points of contact of the tangents drawn from the cMtcrnal point. Any point P lying on the polar of a point Q' has its own polar passing through Q'. The polars of any two points, and the line joining the points, form a triangle called the self-recip- rocal triangle with respect to the circle, the three vertices being the poles of the opposite sides. (4) Involution. — Pairs of inverse points, P, P'; Q> Q'; etc., on the same straight line, form a system in involution, the relation between them being OP OP' = OQOQ' = .... = r'. Here the inverse points are usually called coti- juyate points. Any four points whatever of a system in involution on a straight line have their anharmonic ratio (q.v.) equal to that of their four conjugates. (5) Xine-Points Circle. — The intersection of the three altitudes of a triangle is called the or- thocentre. The mid-points of the segments from the orthocentre to the vertices constitute three points, the feet of the altitudes three more, and the mid-points of the sides of the triangle three more — all nine lying on the circumference of a circle, called the nine-points circle. In Fig. 5, O is the orthocentre and K, L, G, D, M, E, H, N, F are the nine points. DC (6) Seven-Points Circle (Brocard circle). Point S in Fig, 6, so i)laced that its distances from the sides of the triangle ABC are propor- tional to the lengths of the respective sides, is called the synimedian point of the triangle. Lines through this point parallel to the sides cut them in six points. D, E', E, K', F, D', which lie ou a circle called the triplicate-ratio or Fio. 6. Tucker circle. If lines were drawn through A, B, C, parallel to the sides of the triangles DEF, D'E'F', they would intersect one another and F'E, DE', FD' in P, P', L, M, N. These five points,