conics ii
From Casey
342. The isogonal transformation of any line cutting the circumcircle of the triangle is a hyperbola whose asymptotic angle is equal to the angle of intersection of the line and circle.
Dem.ó" Let. ABC be the triangle of reference, and let the line cut the circle in the points D, E\ join AD, AE, then the isogonal conjugates of D, E are the points at infinity on the symetriques of AD, AE with respect to the bisector of the angle IB A 0. Hence the curve is a hyperbola whose asymptotes are parallel to the symetriques of AD, AE ï but the angle between the symetriques of AD, AE is equal to DAE, and there- fore equal to the angle of intersection of DE with the circle. Cor. 1.ó The transformation of any diameter of the circum- circle is an equilateral hyperbola. Hence, to find the equation of an equilateral hyperbola circumscribing the triangle of refer- ence, and passing through any point P, we find the equation of the diameter of the circle which passes through the isogonal conjugate of P, and transform. Thus, the equilateral hyperbola which circumscribes the triangle of reference, and passes through its incentre is (cos £ - cos 0)1 a + (cos C - cos^)//? + (cos A - cos J?)/y = 0. (906) The centre of this hyperbola is the point of contact of the nino- points circle with the incircle of the triangle. Corresponding properties hold for the hyperbolae through the excentres.
Cor. 2.ó The isogonal transformation of any tangent to the circumcircle is a parabola.
Cor. 3.ó The transformation of any line which does not meet the circumcircle in real points is an ellipse.
Cor. 4.ó The transformation of all lines equally distant from the centre are similar conics.
Cor. 5.ó If a conic and a line be isogonal conjugates, their poles, with respect to the triangle, are isogonal conjugates.
For, let the conic and line be Iyz + mzx + nxy = 0, and lx + my + nz = 0, their poles are (l, m, n), (I/l, 1/m, l/n).
NEUBERG'S HYPERBOLAE.
343. The isogonal transformation of the directrices of the Brocard Ellipse, ß 335, Cor. 2, are
cos B sin C/x + cos C sin A/y + cos A sin B/z = 0, (907)
sin B cos C/x + sin C cos A/y + sin A cos B/z = 0. (908)
I have named these conics after M. Neuberg, who first studied their properties. I reproduce here his investigation from Mathesis, tome vi., pp. 5-7.
" If from a point P perpendiculars be drawn to the sides of a triangle ABC, and produced so that the perpendicular on a meets a in AI, b in A2, c in A3
,, b meets b in B1, c in B2, a in B3
,, c meets c in (7^ a in <72, # in (73.
Then, T1, T2, T3 denoting the areas of the triangles A1B1C1
AJBi (72, ^3I?3 (73, respectively. The loci of P, when the triangles T2, T3 vanish, are the hyperbolae (907), (908).
locus of points for which T2 = T3 is
7) + 7asin(tf-^ + 0/3 sin(^--S) ==0. (909)
Kiepert's Hyperbola.
T1:T2+T3 in a constant ratio.
poles with respect to the triangle of reference Neuberg's hyperbolae, Kiepert's hyperbola and circumcircle
idL their line of collinearity is parallel to
x cos A + y cos B + z cos C = 0.
353. Brocard's parabolae
If two isogonal lines y - kz = 0, ky - z =0 meet the altitudes 'Jig., ß 347^ JW21, Cfl"e, f?i in points Q, R the envelope of QR is a parabola.
4x cos A (y cos B + z cos C-x cos A)- (y cos C-z cos B)^2 = 0.
(918)
Cor. 2.óBy giving special values to k we get special positions of QR in each of which it will be a tangent. Thus, if k = 0 we get x = 0 as the tangent, if k = oo ,
y cosB + z cos C- x cos A = 0,
that is, the line HbHc is a tangent, and by making k = ± 1, we see that the internal and external bisectors of the angle BAC are tangents.
354. If P be the point which divides ASa in the same ratio as R divides CH^ the envelopes of the lines RP, PQ will be two other parabolse whose equations are obtained from (918) by interchange of letters, viz.,
4y cos B (z cos C + x cos A - y cosB) = (z cosA + x cos C)2.
(919)
4y cos C(acosA + /3 cos B - y cos C) = (a cos B + /3 cos Af.
(920)
355. The symmedian lines AK, BK, CK are the directrices of the three parabola and the points a1, b1, C1 are their foci.
AKTZT'S PASABOL2E (Second Group).
356. These touch the perpendicular bisectors of two sides of the triangle of reference, and the internal and external bisectors of the angle formed by these sides. Their equations are
{(y sin C+ z sin B) cos A - x sin A)2= sin3A sin (B- C) (z2-y2).
(921)
1. The foci of the Brocard' s parabolas are the isogonal conjugates of the summits of Brocard's second triangle.
The polars of orthocentre .ffare the radii OA, OB, 00 of circumcircle. 2. The foci of Artzt's parabolas are the summits of Brocard's second triangle.
5. The directrices of Artzt's parabolas are the medians of ABC.
6. The side bc of the triangle abc is a tangent to the parahola a version
7. The co-ordinates of the point a are 1/2 tan A, sin B, sin C, (928)
THE BROCARD ELLIPSE.
335. To find the trilinear equation of the Brooard ellipse.
Let S be the centre of inversion, A'B'C' the equilateral triangle of whose summits the points A, B, 0 are the inverses, £' the middle point of ArCr. Join SD' intersecting AC in D. Then (ß 323) DJLs the point of contact of AG with the Brocard Ellipse. Now, in the triangle SA'Cf, AC is antiparallel to A'C', and SD' is the median of SA'G'. Hence, 3D is the symmedian of SAC .-. AD : X>C : : SAZ: SO2. Again, from the pairs of similar triangles, SAB, SB'A', SCB, SB'C' we have SAiABu SB' : B'A'; SC : CB : : SB': B'C', but B'A' = B'C'. Hence SA:A£::SC: CB. Therefore AB*: BC*:: AD]: DC. Hence (J), AB}jAB = (D . BC)/B C.
Therefore, if a, /3, y be the equations of the sides of the tri- angle ABC, and a, b, G their lengths, the equation of BD is z/c - x/a = 0. Hence the equation of the Brocard ellipse is root(x/aa) +...
Cor. 1.ó The reciprocal of the Erocard ellipse with respect to the conic xx/aa + yy/bb + zz/cc = 0 is l/aa is the Steiner ellipse.
Cor. 2.óThe directrices of the Brocard ellipse are
sin B cos C x+ sin C cos A y. + sin A . cos B . z = 0, (896)
cos B sin C x+ cos C sin A y + cos A sin B . z = 0, (897)