Question 146
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 $\angle A$, $\angle B$, 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 $PP^*$ contains $Q$. Consider:
a. the M'Cay cubic $\mathfrak{K}(O)$, where $O$ is the circumcenter of $\triangle ABC$,
b. the Thomson cubic $\mathfrak{K}(G)$, where $G$ is the centroid of $\triangle ABC$,
c. the Napoleon-Feuerbach cubic $\mathfrak{K}(N)$, where $N$ is the nine-point center of $\triangle ABC$,
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 $\mathfrak{K}(X_{30})$, where $X_{30}$ is the point at infinity along line $OG$,
f. the nine-point circle of $\triangle ABC$,
g. the incircle of $\triangle ABC$, and
h. the circumcircle of $\triangle ABC$.
Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E| / 6}\right\rfloor$ points.
Given a point $Q$, let $\mathfrak{K}(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $PP^*$ contains $Q$. Consider:
a. the M'Cay cubic $\mathfrak{K}(O)$, where $O$ is the circumcenter of $\triangle ABC$,
b. the Thomson cubic $\mathfrak{K}(G)$, where $G$ is the centroid of $\triangle ABC$,
c. the Napoleon-Feuerbach cubic $\mathfrak{K}(N)$, where $N$ is the nine-point center of $\triangle ABC$,
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 $\mathfrak{K}(X_{30})$, where $X_{30}$ is the point at infinity along line $OG$,
f. the nine-point circle of $\triangle ABC$,
g. the incircle of $\triangle ABC$, and
h. the circumcircle of $\triangle ABC$.
Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E| / 6}\right\rfloor$ points.