Suppose we're given a triangle $ABC$. At which interior point $T$ is the product of distances $|AT|\cdot |BT|\cdot |CT|$ maximal? Is it a known point, like the centroid or incenter?
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Such a point $T$ doesn't exist!
Identify the Euclidean plane $\mathbb{R}^2$ with complex plane $\mathbb{C}$. Let $t, a, b, c \in \mathbb{C}$ correspond to $T, A, B, C$ respectively. We have
$$|AT||BT||CT| = |t-a||t-b||t-c| = |(t-a)(t-b)(t-c)|$$
As a function of $t$, the RHS is the modulus of a non-constant entire function on $\mathbb{C}$. By maximum modulus principle, it cannot exhibit a true local maximum anywhere on $\mathbb{C}$.
In language of geometry on $\mathbb{R}^2$, there is no point $T$ which locally maximize the expression $|AT||BT||CT|$.
The closest thing one can have are two saddle points (counting multiplicity) corresponds to the two roots of the quadratic polynomial:
$$\frac{d}{dt}\left((t-a)(t-b)(t-c)\right) = 3t^2 - 2(a+b+c)t + (ab+bc+ca) = 0$$
Marden's theorem tell us these two roots are the foci of the Steiner inellipse which is the unique ellipse tangent to the midpoints of the triangle $ABC$.
achille hui
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