This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1890 Excerpt: ...Show also that the same constant and pole of inversion being chosen, the new circle inverts into itself. Ex. 485.--A given straight line is inverted into a circle, the point P' on the circle being the inverse of the point P' on the straight line. Show that another circle can be described touching the given straight ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1890 Excerpt: ...Show also that the same constant and pole of inversion being chosen, the new circle inverts into itself. Ex. 485.--A given straight line is inverted into a circle, the point P' on the circle being the inverse of the point P' on the straight line. Show that another circle can be described touching the given straight line at P and the circle at P'. Show also that the same constant and pole of inversion being chosen, the new circle inverts into itself. PROPOSITION. The inverse of a circle with respect to an external or internal point is another circle. Let O be a point external to the (c) APB. Let AB be the ends of the diamr. which, when produced, passes through O. Take A', B', P', inverse pts. of A, B, P. Then '.' OP.OP' = OA.OA', .-. A, P, P', A' are concyclic OA'P'=aOPA. Simy., A'B'P'=z-BPP'. The above demonstration and diagrams might require some slight modifications if the value of the constant of inversion or the position of the pole were changed. Note that if P be on the part of circle APB which is convex to O, P' will be on the part of circle A'P'B' which is concave to O, and vice versd. Ex. 486.--If OQ be a tangent from O to circle APB, and Q' the inverse of Q, then OQ' is a tangent to circle A'P'B'. Prove (1) with, (a) without, the use of the 'Method of Limits.' Ex. 487.--Two circles being given, a pole and constant of inversion can be found such that each is the inverse of the other. The pole will in general be the cross of either the external or the internal common tangents. Ex. 488.--If p, p' are the other pts. of section of OPP' with the (c)s APB, A'P'B' respectively (diagram above), Show that (1) AP B'p' Ap B'p' BP A'p' Bp A'p' (2) tangts. at P, p are respectively to tangents p', P', (3) segt. PQp is similar to segt. P'Q'p'. (L O PA = L O
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