Hogg. — On Steiner's Envelope. 345 



Again. BGX = i {A + x) = PHN 

 = BPD 



= 90° - (B + x) 

 :. A + X = 180 - 2 (B + x) 



3x = 180° -2B-A=C-B 

 x^^ (C - B). 



Hence the position of the point P is found. 



If K be the orthocentre of the triangle ABC, then by a well-known 

 theorem PK is bisected by DEF ; DEF is parallel to OP ; therefore DEF 

 bisects OK — i.e., it passes through the nine-point centre. Hence since DEF 

 is a tangent to the hypocycloid and passes through the centre of the curve, 

 it is a cuspidal tangent. 



Since the cuspidal tangents meet at angles of 120°, the other points Q 

 and R in which the cubic C meets the circle S and whose pedal lines are 

 cuspidal tangents will form with P an equilateral triangle inscribed in the 

 circle ABC. This follows from the fact that the pedal lines of the extremi- 

 ties of a chord of a circle meet at an angle equal to the angle at which the 

 chord cuts the circle. 



The trilinear ratios of the points P, Q, R are respectively : — 



[cosec X, cosec (C — x), — cosec (B + x)], 



[cosec (C — y), cosec y, — cosec (A + y)]. 



[— cosec (B + z), cosec (A — z), cosec z], 



where 2y and 2z are the angles subtended by QB and RC respectively at A. 



The equations of the cuspidal tangents are 



aa tan x — b/3 tan {G — x) + cy tan (B -{- x) = o 

 — aa tan (C — ?/) -|- bft tan y -\- cy tan (A — y) = o 

 aa tan (B + ^) — b(3 tan (A — 5:) + cy tan 2 = 0. 



The following properties of the points P, Q, R may be noticed : — 



i. The tangent to the circle ABC at P is the axis of the parabola inscribed 

 in the triangle ABC and having its focus at P. 



ii. The rectangular hyperbola which is the isogonal transformation of 

 OP has its asymptotes parallel and perpendicular to OP. 



iii. If P' be the other extremity of the diameter through P, then the 

 pedal line of the point P' will touch the nine-point circle of the triangle 

 ABC. 



iv. If the lines PA, PB, PC meet BC, CA, AB in A'B'C respectively, 

 then the triangle A'B'C is self -conjugate with respect to the parabola, which 

 is the isogonal transformation of the tangent to the circle ABC at P. 



V. The asymptotes of the cubic C = are parallel to the tangents to 

 the circle ABC at P, Q, R, and are concurrent at the centroid of the trianele 

 ABC. 



