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Extra resources for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.
For example, if the circle O v r x is external and the remaining circles internal to 0, R the relation is written 23 n'+Sr M+iT 5^=o, with analogous expressions for all other cases. NOTE. The student must carefully observe that of the three terms of the equation two are positive and one negative ; the latter corresponding to the pairs of circles whose contacts are alternate. Thus in the figure, O v rx r3 have alternate contacts with the given circle, therefore the term 31 24 is negative, and taking the and 3, .
Having given the vertical angle and radius of the in- or corresponding ex-circle, the envelope of the circum-circle is a circle. [By Ex. ] conclude the present chapter with the following useful property, of the common tangents to four circles which touch a fifth, due to the late Dr. Casey. We 7. shall Denote the circle whose centre is and radius r by and let the four circles 0^, 2r2 r touch r 4 4 3 3 a fifth 0, R at the points A, B> C, D. Let the distance cor2 3 be EXAMPLES. The envelope 1. (Euc. III. of equal chords in a circle is a concentric circle 14). If two sides of a given triangle touch 2. Bobillier's Theorem. fixed circles the third side also touches, or envelopes, a circle. [Let ABC be A' and B' and C'0 B 2 2, the centres draw parallels to the sides meeting the base in each other in C'. Describe a circle O^C', and draw parallel to Since Through O l and the given triangle. of the given circles, C'03 A B. is a given angle ( = A\ 3 is a fixed point.
EXAMPLES. The envelope 1. (Euc. III. of equal chords in a circle is a concentric circle 14). If two sides of a given triangle touch 2. Bobillier's Theorem. fixed circles the third side also touches, or envelopes, a circle. [Let ABC be A' and B' and C'0 B 2 2, the centres draw parallels to the sides meeting the base in each other in C'. Describe a circle O^C', and draw parallel to Since Through O l and the given triangle. of the given circles, C'03 A B. is a given angle ( = A\ 3 is a fixed point.