A first course in projective geometry . xis do not cut the circles ortho-gonally. The student should examine the particular cases which thepreceding theorems assume when one or more of the circles COAXAL CIRCLES 55 has zero radius— shrinks up into a point—and, in parti-cular, should work the following exercises : (1) Construct the radical axis of a circle and a point outside it. (2) Describe a circle to cut three circles orthogonally. (3) Describe a circle to cut two circles orthogonally, and to passthrough a given point outside both. (4) Describe a circle to cut a given circle orthogonall
A first course in projective geometry . xis do not cut the circles ortho-gonally. The student should examine the particular cases which thepreceding theorems assume when one or more of the circles COAXAL CIRCLES 55 has zero radius— shrinks up into a point—and, in parti-cular, should work the following exercises : (1) Construct the radical axis of a circle and a point outside it. (2) Describe a circle to cut three circles orthogonally. (3) Describe a circle to cut two circles orthogonally, and to passthrough a given point outside both. (4) Describe a circle to cut a given circle orthogonally, and to passthrough two given points outside it. § 17. Def. Circles which have a common radical axis aresaid to be coaxal or to form a coaxal system. Since theradical axis is perpendicular to the line joining the centres ofany two of the circles, all the circles must have their centreson the same straight line. Construction of circles of the system. Let the radical axismeet the line of centres at X (Fig. 26). Then a circle whose. Fig. 26. centre is X and radius equal to the tangent from X to any onecircle of the system cuts all the circles orthogonally. AVe can construct as many circles of the system as weplease by drawing tangents to this circle from points on the 56 PROJECTIVE GEOMETRY line of centres. The circle whose centre is the point selected,and whose radius is the length of the tangent, is clearly acircle of the system. §18. Prop. Through any given point one and one, onecircle can be drawn coaxal with two given circles. Let X be the foot of the radical axis of the given circles(Fig. 26), T the point through which the required circle is tobe drawn. Draw the circle, centre X, which cuts the system orthogo-nally. The required circle will be cut orthogonally also. It will therefore cut TX at T, the inverse of T with respectto the circle whose centre is X. The required circle passes through T and T, and has itscentre on the given line of centres. It can therefore be
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
