MathDB

36

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(1)

2016 Guts #36

Source:

12/24/2016
<spanclass=′latex−bold′>(Self−IsogonalCubics)</span><span class='latex-bold'>(Self-Isogonal Cubics)</span> Let ABCABC be a triangle with AB=2AB = 2, AC=3AC = 3, BC=4BC = 4. The \emph{isogonal conjugate} of a point PP, denoted P∗P^\ast, is the point obtained by intersecting the reflection of lines PAPA, PBPB, PCPC across the angle bisectors of ∠A\angle A, ∠B\angle B, and ∠C\angle C, respectively.
Given a point QQ, let K(Q)\mathfrak K(Q) denote the unique cubic plane curve which passes through all points PP such that line PP∗PP^\ast contains QQ. Consider:
[*] the M'Cay cubic K(O)\mathfrak K(O), where OO is the circumcenter of △ABC\triangle ABC, [*] the Thomson cubic K(G)\mathfrak K(G), where GG is the centroid of △ABC\triangle ABC, [*] the Napoleon-Feurerbach cubic K(N)\mathfrak K(N), where NN is the nine-point center of △ABC\triangle ABC, [*] the Darboux cubic K(L)\mathfrak K(L), where LL is the de Longchamps point (the reflection of the orthocenter across point OO), [*] the Neuberg cubic K(X30)\mathfrak K(X_{30}), where X30X_{30} is the point at infinity along line OGOG, [*] the nine-point circle of △ABC\triangle ABC, [*] the incircle of △ABC\triangle ABC, and [*] the circumcircle of △ABC\triangle ABC.
Estimate NN, the number of points lying on at least two of these eight curves. An estimate of EE earns ⌊20⋅2−∣N−E∣/6⌋\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor points.