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Art. L.—*On the Inscribed Parabola*

[*Read before the Philosophical Institute of Canterbury, 1st September, 1915*]

1. If *l + m + n = o*, the conic inscribed in the triangle of reference ABC, whose equation is

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S ≡ √ *lax* + √ *mby* + √*ncz* = 0,

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is a parabola whose focus F has trilinear ratios (*a/l, b/m, c/n*).

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It may be remarked that if a line cutting BC, CA, AB in D, E, F respectively move so that the ratio DE DF = *p: q*, the envelope of the line is the parabola √ (*p − q*) + √ − *pb*ॆ + √ *qcz* = 0, *i.e., l; m: n = p − q − p . q*.

The equation of the directrix of the parabola is *l* cos Ax + *m* cos *By + n* cos Cz = 0

The trilinear polar of F and the directrix of S meet on the conic which is the isogonal transformation of Euler's line (*i.e*, the line passing through the orthocentre, centroid and circumcentre)

Let S touch BC, CA, AB in A′, B′, C′ respectively: the equations of B'C′, C'A′, A'B′ are respectively

L ≡ −*lax + mby + ncz = 0*

M ≡ *lax − mby + ncz = 0*

N ≡ *lax + mby − ncz = 0*

The lines L, M, N pass through the fixed points (−1/a, 1/b, 1/c), (1/a, − 1/b, 1/c), (1/a, 1/b, − 1/c) respectively, and the triangles ABC, A'B′C′ are in perspective, the trilinear ratios of P—the centre of perspective—being (1/la, 1/mb, 1/nc). The locus of P is the Steiner ellipse of the triangle ABC (*i.e*, the circumscribed ellipse whose centre is at the centroid, or maximum circumscribed ellipse) The line FP passes through the fixed point, [1/a(b^{2} − c^{2},1/b(c^{2} − a^{2}, 1/c(a^{2} − b^{2})], which is a point of intersection of the circumcircle and Steiner ellipse of the triangle ABC.

The equation of the maximum inscribed ellipse of the triangle A'B′C′ is *l*√ L + m √M + n √ N = 0, hence P is also situated on this conic.

The bisector of the diagonals of the quadrilateral formed by the lines L, M, N and the trilinear polar of P—viz., *lax + mby + ncz = o*—is the line *l*^{2}*ax* + *m*^{2} by + *n*^{2}*cz* = o, which touches its envelope—the Steiner ellipse—at the point P.

The triangle A'B′C′ is self-conjugate with respect to the Steiner ellipse, whose equation may be written *l*L^{2}+*m*M^{2} + *n*N^{2} = 0.

The polar of the focus F (*a/l, b/m, c/n*) with respect to the triangle A'B′C′ has for its equation

L/b^{2} + c^{2} − a^{2} + M/c^{2} + a^{2} − b^{2} + N/a^{2} + b^{2} − c^{2} =0;

*i e.*, tan AL + tan BM + tan CN = *0*,

which may be written in the form

*m* {(tan B + tan C - tan A) *ax* - (tan C + tan A - tan B) *by*} + *n* {(tan B + tan C - tan A) *ax* - (tan A + tan B - tan C) *cz*} = 0, showing that the polar passe through a fixed point.

The sides of the medial triangle of the triangle A'B′G′ envelop parabolas as F moves on the circle ABC. The equation of the side of the medial triangle parallel to B'C′ is − *l*^{2}L + *m*^{2}M + *n*^{2}N = 0, which reduces to

(*m*^{2} + *mn* + *n*^{2}) *ax* + *m*^{2}*by* + *n*^{2}*cz* = 0,

and the envelope of this line is the parabola

*a*^{2}*x*^{2} = 4 (*ax + by) (ax + cz)*.

If S be expanded and − *l (m + n), − m (n + l), − n (l + m)* be substituted for *l*^{2}, *m*^{2}, *n*^{2}respectively, we have

S ≡ *mn (by + cz)*^{2} + *nl (cz + ax)*^{2} + *lm (ax + by)*^{2} = 0,

showing that the triangle whose medial triangle is the triangle ABC is self-conjugate with respect to S.

2. The line at infinity expressed in terms of L, M, N is *l*^{2} L + *m*^{2}M + *n*^{2}N = 0: if (*x*′*y*′*z*′) be the trilinear ratios of the centroid G′ of the triangle A'B′C′, then the polar of G′ with respect to the triangle—viz., L/L′ + M/M′ + N/N′ = 0— is identical with *l*^{2}L + *m*^{2}M + *n*^{2}N = 0. Hence *l*^{2}L′ = *m*^{2}M′ = *n*^{2}N′ = κ, and therefore M′ N′ 2*lax*′ = κ (1/*m*^{2} + 1/*n*^{2}), whence

*ax*′/*l* (*m*^{2} + *n*^{2}) = *by*′/*m* (*n*^{2} + *l*^{2}) = *cz*′/*n* (*l*^{2} + *m*^{2})

The isogonal conjugate of F—viz., the point (*l/a, m/b, n/c*—lies at infinity in a direction perpendicular to the pedal line of F, and therefore parallel to the axis of S. The line joining G′ to the isogonal conjugate of F is (*m − n) ax + (n − l) by + (l − m) cz = 0*, which passes through the centroid G (1/a, 1/b, 1/c) of the triangle ABC. Hence GG′ is parallel to the axis of the parabola.

From the above equations for the centroid G′ *ax*′ = κ *l*^{3} − 2*lmn*), *by*′ = κ (*m*^{3} − 2*lmn*, *cz*′ = κ (*n*^{3} − 2*lmn*), and therefore *ax*′ + *by*′ + *cz*′ = − 3κ *lmn*, and thence we obtain *ax*′ − 2*by*′ − 2*cz*′ = 3κ *l*^{3}.

The locus of G′ is therefore the cubic curve (*ax* − 2*by* − 2*cz*)^{⅓} + *by* − 2*cz* − 2*ax*)^{⅓} + *cz* − 2*ax* − 2*by*)^{⅓} = 0.

The equation of the pedal line of F is *lmn* 2R sin A sin B sin C (*ax + by + cz*) = (*mna*^{2} + *nlb*^{2} + *lmc*^{2}) (*l* cos Ax + *m* cos By + *n* cos Cz); hence, eliminating *l, m, n* between this and the equations, *l + m + n = o, (m - w) ax + (n - l) by + (l - m) cz = o*, we obtain for the pedal of the envelope of the pedal lines of the triangle ABC, the centroid being the pole of the pedal, the equation

2R sin A sin B sin C Σ (*ax*) UVW = Σ (*a*^{2} VW) Σ (cos A *x*U). where U ≡ *by* + *cz* − 2*ax*, V ≡ *cz + ax − 2by*, W ≡ *ax + by − 2cz*.

3. A parabola is the isogonal transformation with respect to any inscribed triangle of a tangent to the circumcircle of the triangle. Hence the parabola S = o, whose equation may be written in the form 1/L + 1/M + 1/N = 0, may be regarded as the isogonal transformation with respect to the triangle LMN of a tangent to the circumcircle of that triangle.

The equation of this circle is

*l*^{2}Ω_{1}/L + *m*^{2}Ω_{2}/M + *n*^{2}Ω_{3}/N = 0,

where Ω_{1}≡ *a*^{2}*l*^{2} + *b*^{2} *m*^{2} + *c*^{2}*n*^{2} − 2*bcmn* cos A + 2*canl* cos B + 2*cablm* cos C, with similar expressions for Ω^{2} and Ω^{3}^{*}

The tangent to this circle at the point Q, determined on it by the equations *l*′L/*l*^{2}Ω_{1} = *m*′M/*m*^{2}Ω_{2} = *n*′N/*n*^{2}Ω_{3} is *l*′^{2}L/*n*^{2}Ω_{1} + *m*′^{2}M/*m*^{2}Ω_{2} + *n*′^{2}N/*n*^{2}Ω_{3} = 0,

where *l*′ + *m*′ *n*′ = 0.

The isogonal transformation of this line with respect to the triangle LMN is *l*′^{2}/*l*^{2}L + *m*′^{2}/*m*^{2}M + *n*′^{2}/*n*^{2}n = 0, which reduces to 1/L + 1/M + 1/N = 0 if *l*′^{2} *m*′^{2}: *n*^{2} *m*^{2} *n*^{2}. Hence S is the isogonal transformation of the tangent L/Ω_{1} + M/Ω_{2} + N/Ω_{3} = 0, whose point of contact Q is given by the equations L/*l*Ω_{1} = M/*m*Ω_{2} = N/*n*Ω_{3}.

To determine the locus of Q we have κ (M + N) = *m*Ω_{2} + *n*Ω_{3}, whence 2κ *lax* = (*m + n*) *l*^{2}*a*^{2} + *m*^{2}*b*^{2} + *n*^{2}*c*^{2} + 2*mnbc* cos A) + 2*l* (*m*^{2}*ab* cos C + *n*^{2}*ca* cos B) − 2*lmn*_{2}, 2κ *ax* = − (*l*^{2}*a*^{2} + *m*^{2}*b*^{2} + *n*^{2}*c*^{2} + 2*mnbc* cos A) + 2 (*m*^{2}*ab* cos C + *n*^{2}*ca* cos B) − 2*mna*_{2} = *l(m + n) a*^{2} + (n + l) b^{2} + n (l + m) c^{2} − 2*mnbc* cos A − 2*m (n + l) ab* cos C − 2*n (l + m) ca* cos B −2*mna*^{2} = − 3*mna*^{2} + *nlb*^{2} + *lmc*^{2}

[Footnote] * See “Messenger of Mathematics,” No. 502, February, 1913, p. 129

Solving for *mn, nl, lm* from the equations

−3*mna*^{2} + *nlb*^{2}*lmc*^{2} = 2κ *ax*

*mna*^{2} − 3*nlb*^{2} + *lmc*^{2} = 2κ *by*

*mna*^{2} + *nlb*^{2} − 3*lmc*^{2} = 2κ *cz*,

we obtain

*a*^{2}*mn*: *b*^{2}*c*^{2}*lm* = 2*ax* + *by* + *cz* : *ax* + 2*by* + *cz*: *ax* + *by* + 2*cz*;

hence the locus of the point of contact Q is

*a*^{2}/2*ax* + *by* + *cz* + *b*^{2}/*ax + 2by + cz* + *c*^{2}/*ax + by + 2cz* = 0,

which on expansion becomes

*abc (ayz + bzx + cxy) + (ax + by + cz)* × [(*a*^{2} + 2*b*^{2} + 2*c*^{2} + 2*a*^{2}) *by* + (*c*^{2} + 2*a*^{2} + 2*b*^{2})*cz*] = 0,

a circle of radius 4R, having its centre at the point whose trilinear ratios are

[cos A - cos B cos C, cos B - cos C cos A, cos C - cos A cos B].

The triangle formed by the lines

*2ax + by + cz = 0, ax + 2by + cz = 0, ax + by + 2cz = 0*

may be called the Δ_{0}.

4. Let S≡ √*lax* + √*mby* + √*ncz* = 0

S′ √*l*′*ax* + √*m*′*by* + √*n*′*cz* = 0,

when *l*′ = *m − n, m′ = n − l, n′ = l − m*, be two parabolas inscribed in the triangle ABC.

The equation of the line joining their foci F(*a/l′, b/m, c/n*), and F′ (*a/l′, b/m′, c/n′*) is *ll′x/a + mm′ y/b + nn′ z/c = 0*,

showing that FF′ is a chord of the circle ABC passing through the symmedian point (*a, b, c*) of the triangle ABC.

The centres of perspective P, P′ of the triangle ABC and the two triangles formed by joining the points of contact of S and S′ with the sides of the triangle ABC have trilinear ratios

1/*la, 1/mb, 1/nc*), (1/*l′a, 1/m′b, 1/n′c*)

respectively: the equation of the line PP′ is *ll′ax + mm′ by + nn′cz* = 0,

which is satisfied by the trilinear ratios of the centroid (*1/a, 1/b, 1/c*), showing that P and P′ are the extremities of a diameter of the Steiner ellipse of the triangle ABC. Hence the lines joining the fourth point of intersection of the circle ABC and the Steiner ellipse of the triangle to the extremities of any chord of the circle, which passes through the symmedian point of the triangle, meet the Steiner ellipse again at the extremities of a diameter of the ellipse.

The polars of the centroid of the triangle ABC with respect to S and S′ form a pair of parallel tangents to the Steiner ellipse of the triangle ABC. The equations of the two polars are *l*^{2}ax + *m*^{2}*by**n*^{2}*cz* = 0, *l′*^{2}*ax* + *m′*^{2}*by* + *n′*.^{2}*cz* = 0, and at their intersection *ax: by: cz = ll′: mm′: mm′ nn′*, showing that the tangents meet on the line at infinity.

The equations of the lines joining P to the isogonal conjugate of F and P′ to the isogonal conjugate of F′ are respectively

*l*^{2}*l′ax* + *m*^{2}*m′by* + *n*^{2}*n′cz* = 0

*ll′*^{2}*ax* + *mm′*^{2}*by* + *nn′*^{2}*cz* = 0

These lines, which are parallel to the axes of S and S′ respectively, intersect on the Steiner ellipse of the triangle ABC in the point 1/*all′*1/*bmm′*, 1/*cnn′*), which is the centre of perspective of the triangle ABC and the “contact” triangle of the inscribed parabola which has its focus at the trilinear pole *a/ll′*, *b/mm′*, *c/nn′*, of the chord FF′.

The equation of the trilinear polar of the isogonal conjugate of F′ is *ax/l′* + *by/m′* *cz/l′* = 0 from its form it is seen that it touches S and also the maximum inscribed ellipse, √*ax + √ by + √ cz* = 0, of the triangle ABC. The corresponding fourth common tangent of this ellipse and the parabola S′ *ax/l + by/m + cz/n* = 0; the trilinear ratios of the point of intersection of these lines are (*ll′/a*, *mm′/b*, *nn′/c*, which satisfy the equation of the line at infinity. Hence these common tangents are parallel; they touch the ellipse at the extremities of the diameter whose equation is *ll′ax* + *mm′by* + *nn′cz* = 0; *i.e.*, the diameter of the Steiner ellipse whose extremities are the centres of perspective P, P′. These common tangents to the maximum inscribed ellipse and the parabolas touch S, S′ at the points *ll*′^{2}/*a*, *mm*′^{2}/*b*, *nn*′^{2}/*c*, (*l*^{2}*l*′/*a*, *n*^{2}*n*′/*b*, *n*^{2}*n*′/*c*, respectively, and the line joining these points—viz., *ax/ll*′ + *by/mm*′ + *cz/nn*′ = 0—is a tangent to the maximum inscribed ellipse.

5. The equation of the lines joining A to the points in which S and S′ touch BC is

(*mby − ncz) (m′by − n′cz)* = 0;

*i.e.*, *mm′ b*^{2}*y*^{2} + *nn*′*c*^{2}*z*^{2}− *(mn′ + m′n) bcyz* = 0,

or *mm*′*b*^{2}*y*^{2} + *nn*′*c*^{2}*z*^{2} − 2*ll*′ *bcyz* = 0.

Hence the six points of contact of S and S′ with the sides of the triangle ABC he on the conic

S^{1} ≡ *ll*′*a*^{2}*x*^{2} + *mm*′*b*^{2}*y*^{2} + *nn*′ + *c*^{2}*z*^{2} − 2*ll*′*bcyz* − 2*cazx*−2*nn*′*abxy* = 0.

If in S_{1} we substitute for (*x, y, z*) the trilinear ratios of the isogonal conjugates of the foci F, F′—viz, *l/a, m/b, n/c*) (*l*′/*a*, *m*′/*b*, *n*′/*c*)—we obtain the expressions *l*^{3}*l*′ + *m*^{3}*m*′ + *n*^{3}*n*′ − 2*lmn* (*l*′ + *m*′ + *n*′),

*ll*′^{3} + *mm*′^{3} + *nn*′^{3} − 2*l*′*m*′*n*′ (*l* + *m* + *n*),

both of which vanish; hence the asymptotes of S_{1} are parallel to the axes of S and S′.

It is at once seen that the conic S_{1} passes through the four fixed points (1/*a*, 1/*b*, 1/*c*), (− 3/*a*, 1/*b*, 1/*c*), (1/*a*, − 3/*a*, 1/*c*, (1/*a*, 1/*b*− 3/*c*).

Its equation may be written

*ll*′/2*ax* + *by + cz* + *mm*′/*ax* + 2*by + cz* + *nn*′/*ax + by + 2cz* = 0.

Hence S_{1}, which is the harmonic conic of S and S′, circumscribes the triangle Δ_{0} It is the locus of points whose polars with respect to Δ_{0} pass through the point (*ll*′/*a*, *mm*′/*b*, *nn*′/*c*,; *i.e.*, whose polars are perpendicular to the pedal line of the point which is the trilinear pole of the chord FF′ of the circle ABC. Its centre lies on the conic

1/*by + cz* + 1/*cz + ax* + 1/*ax + by* = 0.

6. The triangle A'B′C′ is self-conjugate with respect to the conic *l*′/*ax* + *m*′/*by* + *n*′/*cz* =0, whose equation may be written in the form *ll*′ L^{2} + *mm*′ M^{2} + *nn*′ N^{2} =0. This conic passes through the points G 1/*a*, 1/*b*, 1/*c*), and P (1/*al*, 1/*bm*, 1/*cn*), and the tangents to the conic at these points are *l*′*ax* + *m*′*by* + *n*′*cz* = 0 and *l*^{2}*l*′*ax* + *m*^{2}*m*′*by* + *n*^{2}*n*′*cz* = 0 respectively; they are parallel to the axis of the parabola S. The locus of the centre of this conic as F moves round the circle ABC is the maximum inscribed ellipse of the triangle ABC.

7. Let three points F_{1} F_{2}, F_{3}, having trilinear ratios (*a/l*, *b/m*, *c/n*), (*a/m*, *b/n*, *c/l*), (*b/n*, *b/l*, *c/m*) respectively, be taken on the circle ABC. The equation of F^{2}F^{3} is *x/al* + *y/bm* + *z/cn* = 0; hence it touches the conic √*x/a* + √*y/b* + √ *z/c* = 0, which is the Brocard ellipse of the triangle ABC. Hence F_{1}F_{2}F_{3} is a triangle inscribed in the circle ABC and circumscribed to the Brocard ellipse of that triangle.

Let inscribed parabolas S_{1}, S_{2}, S_{3} have their at F_{1}, F_{2}, F_{3} respectively, and let P_{1}, P_{2}, P_{3} be the centres of perspective of the triangle ABC and the “contact” triangles of S_{1}, S_{2}, S_{3} respectively with the sides of the triangle ABC. The trilinear ratios of P_{1}, P_{2}, P_{3} are respectively

(1*la*, 1*nb*, 1*nc*), 1*ma*, 1*nb*, 1*lc*), 1*na*, 1*lb*, 1*mc*), and the equation of P_{2}P_{3} is *ax/l* + *by/m* + *cz/n* = 0, which touches the ellipse √*ax* + √*by* + √*cz* = 0;

hence the centres of perspective determine a triangle inscribed in the Steiner ellipse and circumscribed to the maximum inscribed ellipse of the triangle ABC.

The Steiner ellipse of the triangle P_{1}P_{2}P_{3} is

1/*ax/l* + *by/m* + *cz/n* + 1/*ax/m* + *by/n* + *cz/l* + 1/*ax/n* + *by/l* + *cz/m* = 0,

which reduces to *bcyz* + *cazx* + *abxy* =0; *i.e.*, to the Steiner ellipse of the triangle ABC.

The equations of the parabolas having their foci at F_{2}, F_{3} are, when expanded,

*mnw*^{2} + *nlu*^{2} + *lmv*^{2} = 0

*mnv*^{2} + *nlw*^{2} + *lmu*^{2} = 0,

where *u* ≡ *by + cz, v* ≡ *cz + ax, w* ≡ *ax + by*,

hence *mn*: *nl*: *lm* = *u*^{4} − *w*^{2}*u*^{2}: *w ^{4}* −

*u*

^{2}

*v*

^{2}.

The locus of the intersections of inscribed parabolas which have their foci at the vertices of a triangle inscribed in the circumcircle and circumscribed to the Brocard ellipse of the triangle ABC is 1/*u* + 1/*v* + 1/*w* = 0, where *i* ≡ (*by + cz*)^{4} − (*cz + ax*)^{2} (*ax + by*)^{2}

The condition that an inscribed parabola

S ≡ √*lax* + √*mby* + √*ncz* = 0, *i.e.*, *u*^{2}/*l* + *v*^{2}/*m* + *w*^{2}/*n* = 0,

may pass through the fixed point (*x*_{1}*y*_{1}*z*_{1})is *u*_{1}^{2}/*l* + *v*_{1}^{2}/*m* + *w*_{1}^{2}/*n* =0.. Hence the equation of the two parabolas which may be drawn through the point (*x*_{1}*y*_{1}*z*_{1}) is

1/*v*^{2}*w*_{1}^{2} − *w*^{2}*v*_{1}^{2} + 1/*w*^{2}*u*_{1}^{2} − *u*^{2}*w*_{1}^{2} + 1/*u*^{2}*v*_{1}^{2} − *v*^{2}*u*_{1}^{2} = 0,

and the equation of the two directrices is

*u*_{1}^{2}/*y* cos B − *z* cos C + *v*_{1}^{2}/*z* cos C − *x* cos A + *w*_{1}^{2}/*x* cos A − *y* cos B = 0.

The line joining the foci of the two inscribed parabolas has for its equation *u*_{1}^{2}*x/a* + *v*_{1}^{2}*y/b* + *w*_{1}^{2}*z/c* = 0

If H be any point on the line *fx + gy + hz* = 0, then writing this line in the form *pu + qv + rw* = 0, where *p* ≡ −*f/a + g/b + h/c, q* ≡ *f/a − g/b + h/c*, and *r* ≡ *h/a + g/b − h/c*, we easily find that the line joining the foci of the two inscribed parabolas which pass through the point H envelops the conic *ap*^{2}/*x* + *bq*^{2}/*y* + *cr*^{2}/*z* = 0.

8. If L = 0, M = 0, N = 0 be *any* three lines, the equation of the parabola inscribed in the triangle formed by them is

√*fl*L + √*gm*M + √*hn*N = 0,

where *l + m + n* = 0, and the equation of the line at infinity is

*f*L + *g*M + *h*N = 0.

The equations which determine the focus are

*l*L/*f*Ω_{1} = *m*M/*g*Ω_{2} = *n*N/*h*Ω_{3},

and the equation of the directrix is

*fl*L (− *f*^{2}Ω_{1} + *g*^{2}Ω_{2} + *h*^{2}Ω_{3}) + *gm*M (*f*^{2}Ω_{1} − *g*^{2}Ω_{2} + *h*^{2}Ω_{3}) + *hn*N (*f*^{2}Ω_{1} + *g*^{2}Ω_{2} − *h*^{2}Ω_{3}).

The trilinear ratios of the centre of perspective of the triangle LMN and the triangle formed by joining the points of contact of the parabola with the sides of that triangle are given by the equations

*fl*L = *gm*M = *gm*M = *hn*N.

and these ratios satisfy the equation of the Steiner ellipse of the triangle LMN—viz.,

1/*f*F + 1/*g*M + 1/*h*N = 0.

The trilinear ratios of the isogonal conjugate of the focus is found from the equations

*f*L/*l* = *g*M/*m* = *h*N/*n*,

and the equation of the axis of the parabola is

*f/l*(*g*^{2}Ω_{2}/*m*^{2} − *h*^{2}Ω_{3}/*n*^{2}) L + *g/m* (*h*^{2}Ω_{3}/*n*^{2} − M + *h/n* (*f*^{2}Ω_{1}/*l*^{2} − *g*^{2}Ω_{2}/*m*^{2}) N = 0.

The equations of the sides of the “contact” triangle are

L_{1} ≡ − *fl*L + *gm*M + *hn*N = 0

M_{1} ≡ *fl*L + *gm*M + *hn*N = 0

N_{1} ≡ *fl*L + *gm*M − *hn*N = 0,

and the equation of the parabola may be written

1/L_{1} + 1/M_{1} + 1/N_{1} = 0.