ART. XXXI.-On Isogonal Transformations: Part II. By EVELYN G. HOGG, M.A., Christ's College, Christchurch. [Read before the Philosophical Institute of Canterbury, 4th December, 1907.]

1. IF from a point P perpendiculars PD, PE, PF be drawn to the sides BC, CA, AB respectively of the triangle of reference ABC, it is easily shown that the perpendiculars from A, B, and C on EF, FD, and DE respectively are concurrent in a point P', and that the points P and P' are isogonal conjugates.

If now the point P be supposed to move on to the circle ABC, the point P' will move to infinity, and the pedal triangle DEF will become the Simson line of the point P. Hence we derive the important theorem-"The isogonal conjugate of a point on the circumcircle of the triangle of reference lies at infinity in a direction perpendicular to the Simson line of the given point."

2. In this paper use, will also be made of the following theorem: "The Simson lines of the extremities of a chord of a circle intersect at an angle equal to that at which the chord cuts the circle." This may be easily proved from the consideration that if the perpendicular drawn from any point P on the circle ABC to BC meets that circle again in the point A', then AA' is parallel to the Simson line of P.

3. It has been shown in section 4 of Part I of this paper that the asymptotic angle of the circumconic which is the isogonal transformation of a chord of the circumcircle of the triangle of reference is equal to the angle at which that chord cuts the circle. Combining this with sections 1 and 2 of this paper, we see that the asymptotes of the conic which is the isogonal transformation of a chord PQ of the circle ABC are perpendicular to the Simson lines of the points P and Q.

In general, if S' be the isogonal transformation of a curve S, and if S cut the circle ABC in the points P, Q, R.... then the directions of the asymptotes of S' are perpendicular to the Simson lines of the points P, Q, R

4. If the position of a point P be determined by the intersection of the circle ABC and the conic whose equation is Iẞy + mya + naß = 0,

then the Simson line of the point P is perpendicular to the line la+mẞny = o, and its equation may be written

Q = l2 + m2 + n2 - 2mn cos A-2nl cos B-2lm cos C.

5. The isogonal transformation of a tangent to the circle ABC is a parabola circumscribed to that triangle. If the tangent touch at the point P, then the axis of the parabola is perpendicular to the Simson line of the point P. The isogonal transformations of two tangents TP, TQ are two parabolas having their axes inclined at an angle equal to that at which PQ cuts the circle ABC. Parallel tangents transform isogonally into two parabolas passing through four concyclic points and having their axes mutually perpendicular.

6. Let four points A, B, C, D (no three of which are collinear) be taken, and let the triangles formed by omitting in turn each of the points be called A1, A, A, A, let also the isogonal conjugates of A, B, C, D with regard to the triangles A1, A2, Ag, A, be called respectively A', B', C', D'. If the tangents from A', B', C', D' touch the circumcircles of A1, A2, A3, A in PQ1: PQ2: PQ3: PQ, then the two parabolas which can be drawn through the four given points may be regarded as the isogonal transformation of any pair of tangents to the corresponding circumcircle. Hence we see that the eight points of contact of the tangents may be arranged in two groups of four points such that the Simson lines of the points of each group are parallel to one another. This result may also be expressed by saying that each of the chords of contact PQ cuts its associated circle at the same angle-viz., the angle at which the axes of the parabolas are inclined to each other.

7. If the direction of the axis of a parabola circumscribing the triangle ABC be given, the line of which the parabola is the isogonal transformation may be constructed in the following manner Draw through A a chord AA' perpendicular to the given direction; let the perpendicular from A' on BC meet the circle ABC in the point P, then the tangent at P to the circle ABC will isogonally transform into a parabola whose axis is perpendicular to the Simson line of P, and therefore parallel to the given direction.

let

8. Let a straight line L= la + mẞ + ny = o be taken, and be its distance from the centre of the circle ABC and

the angle at which it cuts that circle. Then if p< R, the radius of the circle ABC, L will isogonally transform into the hyperbola Slẞy + mya + naßo, whose eccentricity (e) is

$
2

given by the relation € = sec From this we may deduce the

following expression for the eccentricity :

We now proceed to the case where p > R. Suppose the line L' = l'a + m'ß + n'y = o be drawn parallel to L and passing through the pole of that line with respect to the circle ABC. Let p' be the distance of L' from the centre of the circle ABC, and let L' cut that circle at the angle '.

The line L transforms into an ellipse, and the angle (4) between its equi-conjugate diameters, expressed in terms of the invariants, ', is given by

Taking S to be 21ẞy + 2mya + 2naß = o, we have

Hence we derive the result that the angle between the equiconjugate diameters of S is equal to the angle at which L' cuts the circle ABC.

Moreover, since L and L' are parallel, their isogonal transformations, S and S', will intersect in four concyclic points : the chords of the circle joining these four points will be equally inclined to the axes of each of the conics in other words, the axes of the two conics will be parallel, and therefore the Simson lines of the two points in which L' cuts the circle ABC

will be perpendicular to the equi-conjugate diameters of the ellipse S. The directions of the axes of that conic may therefore be determined.

If and are the eccentricities of S and S' respectively, we have

Eliminating p and p' by means of the relation pp' = R2, we have

9. The foci of any conic inscribed in the triangle of reference are isogonal conjugates. If the trilinear co-ordinates of one focus be (aß.y.), then the co-ordinates of the other focus K K2 K2 where is the semi-minor axis of the conic. a. Bo Yo

will be

The conic may be regarded as the envelope of a variable line la + mẞ+ny = o, which moves so that the product of the perpendiculars on it from the foci is equal to 2. The relation between l, m, n is easily found to be

and the equation of the inscribed conic is

vaa1 + vẞẞ ̧1⁄2 + √73

=0

If D be the focus (a.ß.y.), then 1, 2, 3 are respectively (DA sin A), (DB sin B), (DC sin C)2.

10. If we take D to be the incentre of the triangle ABC,

In a similar manner the equations of the ex-circles may be at once determined.

Let D be the centre of the circle ABC, then

and we find the equation of the inscribed conic having its foci

at the circum- and ortho-centres of the triangle of reference to be

sin Ava cos A + sin BB cos B + sin Cvy cos C = 0 This conic has the nine-point circle of the triangle as its auxiliary circle, and its eccentricity is

√1-8 cos A cos B cos C

If three conics be inscribed in the triangle of reference (supposed acute), the middle points of the perpendiculars from the vertices on the opposite sides being each a focus of one conic, then the major axes of the conics all pass through the centroid of the triangle.

11. The polar of any point with respect to a rectangular hyperbola self-conjugate with respect to the triangle of reference passes through its isogonal conjugate. Taking the equation of the hyperbola to be la2 + mp2 + ny2 = 0, where l+m+ n = o, the polar of any point P(a'ẞ'y') is la'a 1 1 1

+ mB'B + ny'yo, which passes through P'

Let the polars of P and

a' B P' intersect in P", then the V' W'

m

w'), where U = a (B2 — y2),

V = ẞ (y2 — a3), W = y (a2 – ẞ2). Hence the point P" lies on

a conic which passes through P and P'. Since its equation is independent of l, m, n, we derive the following theorem: Given a fixed triangle and a fixed point, the locus of the intersection of the polars of the given point and its isogonal conjugate with regard to rectangular hyperbolas having a given self-conjugate triangle is a conic passing through the vertices of that triangle, the given point, and its isogonal conjugate.

The tangent at any point of the rectangular hyperbola la2+mp2+ny2 = o passes through its isogonal conjugate. If O be the centre of the hyperbola, and if its asymptotes meet the circle ABC again in the points X, Y, then these points are the isogonal conjugates of the points in which the hyperbola is touched by its asymptotes: hence the diameter XY of the circle ABC will isogonally transform into a rectangular hyperbola whose asymptotes are parallel to those of la2 + m2 + ny2 = 0.

The equation of XY is easily found to be