### Theorem regarding Orthogonal Conics

William Finlayson
1907 Proceedings of the Edinburgh Mathematical Society
DEFINITION. When two conies intersect each other at two points in such a manner that the tangents and normals of the one become the normals and tangents of the other, they may be said to cut each other orthogonally. Theorem. 1st, A given conic can be cut at every point on it by two conies which are orthogonal to it; 2nd, every conic orthogonal to a given conic passes through two fixed points on the axis of the given conic. Figure 4 , be the foci of the given conic, Xx and Yy the directrices,
more » ... let R equal the radius of director circle, centre F, and let P be any point on the curve. Draw PSPj a focal chord; then Sx, at right angles to PSP" cuts the X directrix in x, the centre of the orthogonal circle to F which touches PSP, at S, and, since PSP" FP and FP, are all tangents to this circle, it is therefore the in-circle of triangle PFP,. Now the normals at P and P, bisect the exterior angles of PFP, at P and P, and the bisectors meet on Fa; since Fx bisects angle PFP,. Calling this point H, we observe that it is the centre of an ex-circle to triangle PFP, which touches PSP, in S, and FP in / ' so that, if F / 1 be taken as the radius of the director circle and S, as a focus, we get P as a point on an ellipse whose foci are F and S, and whose tangent and normal at P are the normal and tangent to the given conic at P ; for P / 1 = PS, and therefore FP + S, P = F / 1 = R,. Similarly S, P, + FP, = F / 1 = R,: therefore the given conic is cut orthogonally at P and P, by the ellipse whose foci are F and S, and whose directrix is HX" a line through H at right angles to FS,. Let F and S, The second conic is the Hyperbola whose focal chord in the given conic is FP which cuts the conic in Q and P. Determining Q and taking the normals at Q and P, we can see that they will meet at a point H, on the bisector of the exterior angle at S which is the line yj\$, the point y 2 being the centre of the circle of the Y system which was used to determine Q, and S being the focus through which the focal chord QP does not pass. Using the