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Volume 45, 1912
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– 344 –

Art. XLIV.—On Steiner's Envelope.

[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' (α'β'γ') with respect to the circle, S ≡ αβγ + bγα + cαβ = o, circumscribing the triangle of reference ABC, is α (bγ' + cβ') + β (cα' + αγ') + γ (αβ' + bα') = o. If this line passes through the point (1/α' 1/β' 1/γ'), the isogonal conjugate of O', then the locus of O' is the cubic curve C ≡ α2 (bβ + cγ) + β2 (cγ + aα) + γ2) aα + bβ) = o (i)

If α be eliminated between the equations S = o and C = o, the following cubic is given for finding the ratio β:γ of the intersections of the two curves:— c(c2 — a2) β3 + 3bc2 β2γ + 3b2c βγ2 + b (b2 — a2) γ8 = o (ii)

The functions H and G of this cubic are H =–a2b2c2, G = a2bc2 {b2 (a2 + c2)–(c2–a3)2}, whence G2 + 4H8 =–16 Δ2 a4b2c4 (c2–a2)2, where Δ 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 transformation of this tangent will be a parabola circumscribing the triangle ABC, and passing through the point P. The axis of this parabola will be perpendicular 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 O, 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 G, 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.

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Again, BGX = ½ (A + x) = PHN = BPD = 90°–(B + x) ∴ A + x = 180–2 (B + x) ∴ 3x = 180°–2B–A = C–B ∴ x = ⅓ (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 extremities 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 aα tan x–bβ tan (C–x) + cγ tan (B + x) = o–aα tan (C–y) + bβ tan y + cγ tan (A–y) = o aα tan (B + z)–bβ tan (A–z) + cγ tan z = o.

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' 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 = o are parallel to the tangents to the circle ABC at P, Q, R, and are concurrent at the centroid of the triangle ABC.