A treatise on the geometry of the circle and some extensions by William J. M'Clelland

By William J. M'Clelland

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Extra info for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.

Example text

Therefore if one vertex of a triangle of given and another vertex describe a circle, the vertex and all other points invariably conremaining species is fixed, nected with 1. it likewise describe circles. EXAMPLES. Having given the diagonals and angles A BCD, construct of a quadrilateral it. [On one diagonal A C describe segments of circles containing angles B and D. Let A BCD be the required quadrilateral. Produce CD to Y and BC to X. Join B Y and A Y. respectively equal to Then since the chord BY of a given circle subtends a given angle C The triangle ADYis also given in species length, describe a segment of a On hence the following construction it is of known ; : circle containing an angle C.

Again, since the triangle OPQ one vertex a fixed point; if is P fixed in species and describes a line BC it Q is also a line (CA). And one ivhen vertex of a figure of given species is generally, or point invariably conother vertex and fixed any follows that the locus of P nected with it describe describe loci, which it through a a may locus, the remaining points Q be derived known angle POQ and by revolving increasing or OP in the ratio of OQ OP. thus described are similar, the ratio OP termed their Ratio of Similitude and the point diminishing The loci ...

Similarly [Let OD = OD f ; f and since CD=C'D take the equals there remains the triangles COD and C'OD' are the equal areas A BCD and ABC'D' and C'OD' and the common part A OB, and equal in every respect. From COD BOC+ AOD=BOC' + A OD' t MAXIMUM AND MINIMUM. 39 BOO' - BOC= AOD-AOU, or BO. 00= AO. DO, hence from which Ex. manifest that it is CD AB are parallel. and Cf. Art. 9, 5. A similar proof may be applied to show that when of a quadrilateral are given the area a is the four sides maximum when CO.

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