Wikipedia:Reference desk/Archives/Mathematics/2016 August 30

= August 30 =

Locus of P so that Angle PIP` = Given Constant
Let P and P` be isogonal conjugates with respect to the incenter I of triangle ABC, and &alpha; a given constant. What is the locus of P so that PIP` = &alpha; ? (In the case of an equilateral triangle, the locus seems to be a group of 6 hyperbolas. Does the same hold true for all other types of triangles as well ?) — 79.113.241.113 (talk) 18:32, 30 August 2016 (UTC)


 * This may not be much help, but here are a couple random thoughts. In any triangle, if P is on an angle bisector, P' will be on the same angle bisector but on the other side of the incenter, giving angle PIP' = 180°. So the locus for &alpha; = 180° is the union of the three angle bisectors. So loosely extrapolating, at least for large angles the locus will be something whose degenerate case is three intersecting lines.


 * A plodding approach you could try starts by noting that the line through P(p, q, r) and I(1, 1, 1) in trilinear coordinates is, by Trilinear coordinates, $$x(q-r)+y(r-p)+z(p-q)=0.$$ Now by the isogonal conjugate article, P' is (1/p, 1/q, 1/r ), so the line through I and P' is $$x(1/q-1/r)+y(1/r-1/p)+z(1/p-1/q).$$ We want the angle between these two lines, which can be computed from p, q, r and the cosines of the vertex angles, using the formula in Trilinear coordinates. It gets messy, but maybe you could set this angle equal to a fixed &alpha; and see if you come up with an expression in p, q, r that has the form of a conic section in trilinear coordinates as given in trilinear coordinates.


 * Just out of curiosity, how did you obtain your result about equilateral triangles--theoretically, or using a graphing package? Loraof (talk) 21:34, 30 August 2016 (UTC)


 * The latter. :-) — 79.113.241.113 (talk) 22:16, 30 August 2016 (UTC)


 * I arrive at a highly symmetrical and homogeneous cubic polynomial equation in p, q, r, but unfortunately I am not familiar with the form of the equation of a hyperbola in trilinear coordinates. — 79.118.182.223 (talk) 18:24, 2 September 2016 (UTC)