152 BOOK OF MATHEMATICAL PROBLEMS. 



737. If two points be taken on an ellipse such that the 

 normals at these points intersect a given normal in the same 

 point, the chord joining the two points envelopes a parabola 

 whose directrix passes through the centre and whose focus is 

 the foot of the perpendicular from the centre on a tangent per- 

 pendicular to the given normal. 



738. In an ellipse given the centre and directrix, prove that 

 the envelope is two parabolas having their common focus at the 

 given centre. 



739. In an ellipse given one extremity of the minor axis and 

 a directrix, prove that the envelope is a circle having its centre at 

 the given point, and touching the given straight line. 



740. If three points be taken on an ellipse, such that their 

 centre of gravity is a fixed point, the straight lines joining them 

 two and two will touch a fixed conic. 



741. If three points be taken on an ellipse, such that the 

 centre of perpendiculars of the triangle formed by joining them 

 is a fixed point, the joining lines will touch a fixed conic whose 

 asymptotes are perpendicular respectively to the tangents drawn 

 from the fixed point to the ellipse. 



t 



VIII. Areal Coordinates. 



In this system, the position of a point P with respect to three 

 fixed points A, B, C, not in one straight line, is determined by the 

 values of the ratios of the triangles PEC, PC A, PAE to the 

 triangle AC y any one of them PEC being positive or negative, 

 as P and A lie on the same or opposite sides of EC. These are 

 usually denoted by a, /?, y, which satisfy the equation a+fi+y = 1. 

 A point is therefore completely determined by equations of the 

 form la = mp = ny. 



The general equation of a straight line is la + m/3 + ny = 0, and 

 I, 77i, n are proportional to the perpendiculars from A,B y C on the 



