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Omni-MATH / (Self-Isogonal Cubics) Let ABC be a triangle with AB=2,AC=3,BC=4. The isogonal conjugate of a point P, denoted P^{*}, is the point obtained by intersecting the reflection of lines …

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(Self-Isogonal Cubics) Let ABCA B C be a triangle with AB=2,AC=3,BC=4A B=2, A C=3, B C=4. The isogonal conjugate of a point PP, denoted $P^{*}$, is the point obtained by intersecting the reflection of lines PAP A, PB,PCP B, P C across the angle bisectors of ∠A,∠B\angle A, \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 $P P^{*}$ contains QQ. Consider: (a) the M'Cay cubic K(O)\mathfrak{K}(O), where OO is the circumcenter of △ABC\triangle A B C, (b) the Thomson cubic K(G)\mathfrak{K}(G), where GG is the centroid of △ABC\triangle A B C, (c) the Napoleon-Feurerbach cubic K(N)\mathfrak{K}(N), where NN is the nine-point center of △ABC\triangle A B C, (d) the Darboux cubic K(L)\mathfrak{K}(L), where LL is the de Longchamps point (the reflection of the orthocenter across point O)O) (e) the Neuberg cubic K(X30)\mathfrak{K}\left(X_{30}\right), where X30X_{30} is the point at infinity along line OGO G, (f) the nine-point circle of △ABC\triangle A B C, (g) the incircle of △ABC\triangle A B C, and (h) the circumcircle of △ABC\triangle A B C. Estimate NN, the number of points lying on at least two of these eight curves.
Plain-text mathematical notation (without MathML)
(Self-Isogonal Cubics) Let ABC be a triangle with AB=2,AC=3,BC=4. The isogonal conjugate of a point P, denoted $P^{*}$, is the point obtained by intersecting the reflection of lines PA, PB,PC across the angle bisectors of ∠A,∠B, and ∠C, respectively. Given a point Q, let K(Q) denote the unique cubic plane curve which passes through all points P such that line $P P^{*}$ contains Q. Consider: (a) the M'Cay cubic K(O), where O is the circumcenter of △ABC, (b) the Thomson cubic K(G), where G is the centroid of △ABC, (c) the Napoleon-Feurerbach cubic K(N), where N is the nine-point center of △ABC, (d) the Darboux cubic K(L), where L is the de Longchamps point (the reflection of the orthocenter across point O) (e) the Neuberg cubic K(X₃₀), where X₃₀ is the point at infinity along line OG, (f) the nine-point circle of △ABC, (g) the incircle of △ABC, and (h) the circumcircle of △ABC. Estimate N, the number of points lying on at least two of these eight curves.
Original LaTeX notation
(Self-Isogonal Cubics) Let $A B C$ be a triangle with $A B=2, A C=3, B C=4$. The isogonal conjugate of a point $P$, denoted $P^{*}$, is the point obtained by intersecting the reflection of lines $P A$, $P B, P C$ across the angle bisectors of $\angle A, \angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak{K}(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $P P^{*}$ contains $Q$. Consider: (a) the M'Cay cubic $\mathfrak{K}(O)$, where $O$ is the circumcenter of $\triangle A B C$, (b) the Thomson cubic $\mathfrak{K}(G)$, where $G$ is the centroid of $\triangle A B C$, (c) the Napoleon-Feurerbach cubic $\mathfrak{K}(N)$, where $N$ is the nine-point center of $\triangle A B C$, (d) the Darboux cubic $\mathfrak{K}(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O)$ (e) the Neuberg cubic $\mathfrak{K}\left(X_{30}\right)$, where $X_{30}$ is the point at infinity along line $O G$, (f) the nine-point circle of $\triangle A B C$, (g) the incircle of $\triangle A B C$, and (h) the circumcircle of $\triangle A B C$. Estimate $N$, the number of points lying on at least two of these eight curves.

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task_id
06190a0a-f16f-565b-8e61-fe40611d6dd6
task_key
test--06190a0a-f16f-565b-8e61-fe40611d6dd6
task_revision_id
1
config / split
/ test

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