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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 α a given constant. What is the locus of P so that PIP` = α ? (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 α = 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#Collinearities and concurrencies, $x(q-r)+y(r-p)+z(p-q)=0.$ 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#Angle between two lines. It gets messy, but maybe you could set this angle equal to a fixed α 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#Quadratic curves.

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)