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Volume 41, 1908
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Art. VIII.—On certain Conic-loci of Isogonal Conjugates.

[Read before the Philosophical Institute, of Canterbury, 1st July, 1908.]

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1. The locus of a point P (αβγ) which moves so that the line joining it to its isogonal conjugate P' 1/α 1/β 3/γ) passes through a fixed point α0β0γ0) is the cubic α/α022)β/β022)+γ/γ022)=0,

If, however, the point (α0β0γ0) lie on either the internal or external bisector of an angle of the triangle of reference, the cubic becomes a conic and a straight line; and the object of the present paper is to investigate certain properties which this family of conics possesses. For the sake of brevity the fixed point (α0β0γ0) through which the line joining any point to its isogonal conjugate passes will be called the centrum of the conic.

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The co-ordinates of the centrum, are comprised in the system (α0±1±1) if we limit ourselves merely to the internal and external bisectors of the angle A of the triangle of reference ABC. The following four types of conic exist:— centrum011) α2+βγ-α0α(β+γ)=0…I centrum0-11) α2-βγ+α0α(β-γ)=0…II centrum01-1) α2-βγ-α0α(β-γ)=0…III centrum0-1-1) α2+βγ+α0α(β+γ)=0…I

2. These conics possess the following properties: they all pass through the vertices B and C of the triangle of reference; those of classes I and IV pass through the ex-centres I2 and I3; those of classes II and III pass through the in-centre I and the ex-centre I1. The tangents to the conics at I and I1 or at I2 and I3 pass through the centrum; the tangents to the conics at B and C meet at the isogonal conjugate of the centrum. Hence, when the position of the centrum has been assigned, the centre of the conic can be constructed geometrically.

Furthermore, the chord of intersection of any conic of this family with the circumcircle of the triangle of reference is parallel to either the internal or external bisector of the angle A of that triangle. Suppose any conic to cut the circle ABC in the points P and Q: then, since the isogonal conjugate of any point on that circle lies at infinity in a direction perpendicular to the Simson line of the point, the isogonal conjugates of P and Q will be at infinity in directions perpendicular to the Simson lines of those points—that is to say, the asymptotic angle of the conic is equal to the angle between the perpendiculars from the centrum on the Simson lines of P and Q.

If the position of the chord of intersection of the conic and circle ABC is determined, the position of the asymptotes, and therefore of the

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axes, may be found, since the centre is known while the eccentricity of the conic may be deduced from the relation e2=2R/p+R where R is the radius of the circumcircle, and p is the length of the perpendicular from the circumcentre on the chord of intersection.

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3. We now proceed to deal with certain particular conics of this family. The line β — γ = o will meet the line at infinity in the point (b + c, — a, — a); with this point as centrum we have the conic a(α2+βγ)+(b+c)α(β+γ)=0…A which may be written αβγ+βγα+cαβ+α(aα+bβ+cγ)=0

It is the circle on I2I3 as diameter. Hence the theorem—

“Any line parallel to the internal bisector of the angle A of the triangle ABC meets the circle described on I2I3 as diameter in two points which are isogonal conjugates with respect to that triangle.”

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The external bisector β + α = o meets the line at infinity in the point (b — c, — a, a). This point being taken as centrum, we have the conic a(α2+βγ)+(b+c)α(β+γ)=0…B which is the circle on the line II, as diameter. Hence the theorem—

“Any line parallel to the external bisector of the angle A of the triangle ABC meets the circle described on II1 as diameter in points which are isogonal conjugates with respect to that triangle.”

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4. The line β — γ = o meets the circle ABC in the point (— a, b + c, b + c): with this centrum we have the conic (b+c)(α2+βγ)+aα(β+γ)=0…C It meets the circle ABC along the line (b+c)(aα-cβ+bγ)+α2(β-γ)=0, which may be written (b+c)(aα+bβ+cγ)-[(b-c)22](β+γ)=0.

This is satisfied by the coordinates of the centre of the circle ABC. Hence the chord of intersection is the diameter parallel to the external bisector of the angle A. Therefore, since the Simson lines of the extremities of a diameter of a circle are at right angles to each other, we see that the conic is a rectangular hyperbola. The tangents to the conic at B and C are parallel to the internal bisector of A, hence the centre of the conic is the middle point of BC.

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The line β + α = o will meet the circle ABC in the point (a, — b + c, b — c); with this centrum we have conic (b-c)(α2-βγ)+aα(β-γ)=0…D.

Its chord of intersection with the circle ABC is the diameter parallel to the internal bisector of the angle A, and its centre is at the middle point of BC

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5. Let D, E, F, be respectively the middle points of BC, CA, and AB: then the equations of EF, FD, and DE are -aα+bβ+cγ=0…(i) aα-bβ+cγ=0…(ii) aα+bβ-cγ=0…(iii)

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The line β — γ = o will meet the first of these lines in the point (b + c, a, a): using this point as centrum we have the conic a(α2+βγ)-(b+c)α(β+γ)=0…E which meets the circle ABC along the line aα-(b+2c)β-(c+2b)γ=0, or aα+bβ+cγ-2(b+c)(β+γ)=0.

This line meets the internal bisector of angle A at the point [3(b + c), a, a], which is the middle point of the line joining A to the point in which the internal bisector of A meets EF. Hence the conic is a hyperbola, whose centre and asymptotes are found in the manner previously employed.

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The line β — γ = o meets the lines (ii) and (iii) respectively in the points [(b — c), a, a] and [— (b — c), a, a]. With these points as centra we obtain the conics a(α2+βγ)-(b+c)α(β+γ)=0…F a(α2+βγ)-(b+c)α(β+γ)=0…G The former of these conics meets the circle along the line aα-bβ-(2b-c)γ=0, which is parallel to β + γ = o and passes through the middle point of AB. The latter conic meets the circle ABC along the line aα-bβ-(2b-c)γ=0, a line parallel to β + γ = o and passing through the middle point of AC. Hence each of the conics F and G is a hyperbola.

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6. The line β + γ = o will meet the lines (i), (ii), and (iii) of the preceding section respectively in the points [(b-c),a,-a],[(b+c),a,-a],[(b+c),-a,a]. Using these points as centra we have the conics a(α2-βγ)-(b-c)α(β-γ)=0…H a(α2-βγ)-(b+c)α(β-γ)=0…J a(α2-βγ)+(b+c)α(β-γ)=0…K Their chords of intersection with the circle ABC are respectively

These lines are all parallel to the internal bisector of the angle A: the first of them passes through the point [3 (b — c), a, — a.], which is the middle point of the line joining A to the point in which the external bisector of A meets EF. The second and third lines pass respectively through the middle points of AB and AC.

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7. If the centrum be taken at the point (011) in which the internal bisector of the angle A meets BC we have the conic α2+βγ=0…L touching AB, AC, at B and C respectively, and passing through I2 and I3.

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This conic meets the circle ABC in the line aα-cβ-bγ=0, showing that it passes through the point in which the tangent to the circle ABC at A meets BC. This chord may also be written in the form aα+bβ+cγ-(b+c)(β+γ)=0,, showing that it is parallel to the external bisector of the angle A. It has real intersections with the circle ABC, hence the conic is a hyperbola.

The tangents at I2 and I3 meet at the centrum: hence the centre is the point in which the median drawn from A meets the line joining the centrum to the middle point of I2I3. The position of the axes of the conic is therefore given.

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8. If the centrum be taken at the point (0-11) in which the external bisector of the angle A meets BC we obtain the conic α2-βγ=0…M which intersects the circle ABC along aα+bβ+cγ-(b-c)(β-γ)=0, a line which passes through the intersection of BC and the tangent to the circle ABC at A. The intersections of this line with the circle ABC are real if a2>4bc, in which case the conic is a hyperbola. If a2 = 4bc, the conic is a parabola whose axis is perpendicular to the Simson. line of the point in which the above line touches the circle ABC. If a24bc, the conic is an ellipse: to determine the direction of its axes, draw through the pole of this line with respect to the circle a line parallel to it, then the equi-conjugate axes of the ellipse are perpendicular to the Simson lines of the two points in which this line cuts the circle ABC, and the directions of the axes are therefore obtained.

9. The conics of this family possess the property that the isogonal transformation of the tangent to a conic at any point P is a conic circum scribing the triangle of reference and touching the given conic at the point P'which is the isogonal conjugate of P.

Hence one of the common tangents to the circles described on I1I2, I1I3 as diameters will be a circumconic which touches each of these circles at the points isogonally conjugate to the points of contact of the common tangent.

10. Let any straight line L meet the circles described on the lines I2I3, I3I1, I1I2 as diameters in the three pairs of points P1P2, P3P4, P5P6 respectively; let chords of the three circles be drawn through P1P2 parallel to the internal bisector of A, through, P3P4 parallel to the internal bisector of B, and through P5P6 parallel to the internal bisector of C; and let the extremities of these six chords be respectively P1'P2', P3'P4', P5'P6'. Further, let the line L cut the three circles on II1, II2, II3 as diameters in the pairs of points Q1Q2, Q3Q4, Q5Q6, and let chords of these circles drawn parallel to the external bisectors of A, B, and C respectively have their extremities at Q1Q2', Q3'Q5, Q6'. Then the fifteen points A, B, C, the six points P', and the six points Q' all lie on the conic which is the isogonal transformation of the line L.

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11. The general form of the conic which is satisfied by the coordinates of a point and of its isogonal conjugate is α(lα+mβ+nγ)±(lβγ+mγα+nαβ)=0, the centrum of the conic being the intersection with lα + mβ + nγ = o of either the internal or the external bisector of the angle A of the triangle of reference.

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The locus of a point moving so that the line joining it to its isogonal conjugate is parallel to the line lα + mβ + nγ = o is the cubic (mc-nb)α(β2-y2)+(na-lc)β(γ22)+(lb-ma)γ(α22)=0, which, if any two of the three quantities mc—nb, na — lc, lb — ma be equal or have their sum zero, reduces to a conic and either the internal or external bisector of an angle of the triangle of reference. If the two latter quantities be equal, the conic is (na-lc)(α2+βγ)-(mc-nb)α(β+γ)=0, which, since a (m + n) = l (b + c), at once reduces to the form I of section 1 of this paper. A similar reduction occurs if the sum of the two quantities in question be zero.