RIGGS: ON pascal's LIMACON AND THE CARDIOID. 



93 



Below we give a number of theorems respecting the cardioid ob- 

 tained by inverting the corresponding theorems concerning the parabola. 



The straight line which bisects The nodal circle which bisects 



the angle contained by two lines the angle between the line drawn 



drawn from the same point in a from any point on a cardioid to 



parabola, the one to the focus, the the cusp and the nodal circle 



the other perpendicular to the through the point which cuts the 



directrix, is a tangent to the para- 

 bola at that point. 



The latus rectum of a parabola 

 is equal to four times the distance 

 from the focus to the vertex. 



If a tangent to a parabola cut 

 the axis produced, the points of 



director circle orthogonally, is a 

 tangent circle at that point. 



The latus rectum of a cardioid 

 is equal to its length on the axis. 



If a nodal tangent circle cut the 

 axis of a cardioid, the points 



contact and of intersection are of intersection and of tangency 



are equally distant from the cusp. 



equally distant from the focus. 



If a perpendicular be drawn 

 from the focus to any tangent to 

 a parabola, the point of intersec- 

 tion will be on the vertical tangent. 



The directrix of a parabola is 

 the locus of the intersection of 

 tangents that cut at right angles. 



The circle described on any 

 focal chord of a parabola as 

 diameter will touch the directrix. 



The locus of a point from which 

 two normals to a parabola can be 

 drawn making complementary 

 angles with the axis, is a parabola. 



Two tangents to a parabola 

 which make equla angles with the 

 9.xis and directrix respectively, 



If a nodal circle be drawn tan- 

 gent to a cardioid, the diameter 

 of such circle passing through the 

 cusp will be a common chord of 

 this circle and another described 

 on the axis of the cardioid as 

 diameter. 



The base circle is the locus 

 of the intersection of nodal circles 

 tangent to a cardioid, which cut 

 orthogonally. 



The circle described an any 

 nodal chord of a cardioid as diam- 

 eter will be tangent to the base 

 circle. 



The locus of the point through 

 which two nodal circles, cutting 

 a cardioid orthogonally, and mak- 

 ing complementai y angles with the 

 axis, can be drawn is a cardioid. 



Two nodal circles tangent to a 

 cardioid which make equal angles 

 with the axis and latug rectum, 



