CONIC SECTIONS, ANALYTICAL. 173 



in b ; and through b, draw straight lines perpendicular to OA, 

 AS, meeting in c ; then the points a, b, c will be the poles of the 

 sid. s of the triangle, and the straight lines be, ca, ab the polars of 

 the points A, B, C, with respect to some circle with centre 0. 

 Now, suppose we want to find the point corresponding to the per- 

 pendicular from A, it must lie on be, and on the line through at 

 right angles to Oa, since Oa is parallel to the straight line whose 

 reciprocal is required; it is then determined. Hence, to the 

 theorem that the three perpendiculars of a triangle meet in a 

 point, corresponds the following : if through any point in the 

 plane of a triangle ale be drawn straight lines at right angles to 

 Oa, Ob, Oc, to meet the respectively opposite sides, the three 

 points so determined will lie in one straight line. 



So, from the theorem, that the bisectors of the angles meet 



in a point, we get the following : the straight lines drawn through 



cting the external angles (or one external and two internal 



angles) between Ob, Oc ; Oc, Oa ; Oa, Ob, respectively, will meet 



the opposite sides in three points lying in a straight line. 



If a circle, with centre A and radius 7?, be reciprocated with 

 respect to 0, the corresponding curve is a conic whose focus is 0, 



OA 2k* 



major axis along OA, eccentricity ', , and latus rectum -^- or 



. 



j, if we take the radius of the auxiliary circle unity. The cen- 



n. .! located into the directrix. Most of the focal pro- 



perties of conies may thus be deduced from well known jn-oj 

 of the circle. For ins: bo a point on the circle, and 



"/'. OQ two chords at right angles, PQ passes through the centre. 

 Reciprocating with respect to 0, to the circle cnvsp.. : 

 parabola, and to the points P, Q two tangents to tin- parabola 



_;ht angles to each other, which therefore intersect on the 



irix. 



Again, to find the < that two conies which have a 



should be such that triangles can be in 

 one whose sides shall touch the oth< r. 



