344 Transactions. 



Art. XLIV. — On Steiner's Envelope. 



By E. G. Hogg, M.A., F.R.A.S., Christ's College, Christchurch. 



[Read before the Philosophical Institute of Canterbury, 4th December, 1912.] 



The envelope of the pedal lines of a triangle is a tri-cusp hypocycloid, whose 

 centre is at the nine-point centre of the triangle. This curve — known as 

 " Steiner's envelope " — has been discussed in many memoirs, but^ so far as 

 the author of the present paper is aware, no determination has yet been 

 made of the three points whose pedal lines are the cuspidal tangents of 

 the hypocycloid, and the object of this note is to show how these points 

 may be found, and to indicate briefly certain interesting properties which 

 they possess. 



The polar of any point O' (a'/3'y') with respect to the circle, S = afty 

 + hya -f- Ca/3 — 0, circumscribing the triangle of reference ABC, is 



a {by' + C^') + /? (Ca' + ay') + y («/5' + ba') = 0. 



If this line passes through the point I — , jy, -, j , the isogonal conjugate 

 of 0', then the locus of 0' is the cubic curve 



C EI a2 (6/3 + Cy) + /32 (cy + aa) + y' {da + b/3) = ... (i) 



If a be eliminated between the equations S = o and C = o, the follow- 

 ing cubic is given for finding the ratio /3 : y of the intersections of the two 

 curves : — 



c{c^ - a^) /S' + Sbc^ ^^y + 3b-'c/3y- + 6 (6- - a^)y' = o ... (ii) 



The functions H and G of tliis cubic are H = — a^b^c^, G = a-bc'^ 

 \b^a^ + c^)-{c'' -a^)""], whence G^ + 4H'" = - 16 A ^ a*b^c' {c~ - a^)\ 

 where A is the area of the triangle ABC. 



Hence it follows that the roots of the cubic (ii) are all real — i.e., the 

 cubic C meets the circle ABC in three real points besides the vertices of the 

 triangle ABC. 



Let P be one of the points of intersection of the circle and cubic C. The 

 tangent to the circle at P will, since it passes through the isogonal conjugate 

 of P, be perpendicular to the pedal line of P, and the isogonal transforma- 

 tion of this tangent will be a parabola circumscribing the triangle ABC, and 

 passing through the point P. The axis of this parabola will be perpen- 

 dicular to the pedal line of P ; it will also be parallel to one of the bisectors 

 of the angles between the common chords of intersection AB, CP of the 

 circle and parabola. 



Let the arc AP subtend the angle 2x at 0, the centre of the circle ABC ; 

 let D, E, F be the feet of the perpendiculars from P on BC, CA, and AB 

 respectively ; let PC meet AB in Gr, and let GX be the bisector of the angle 

 BGC. Let PO and PD meet AB in H and N respectively. 



Since FD and HP are parallel, FDP = x = HPN. Hence in the triangles 

 PNH, PNB, since HPN = PBN, and PNH is common, the two triangles are 

 equiangular. 



