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I came across this property while playing with elementary triangle constructions.
It looks simple and elegant, but I cannot find a reference to it in open sources.
Are there nice problems, applications, or generalizations of it?
Is there a connection with Ceva's theorem?

Let $ABC$ be an arbitrary triangle, and $P$ a point inside it.
Let us draw lines through $P$ parallel to the sides of $ABC$, thus dividing the sides of $ABC$ onto three segments each.
Let us denote the segments in a circular order:

  • from the point $A$ to $B$ as $c_1$, $c_2$, $c_3$,
  • then from $B$ to $C$ as $a_1$, $a_2$, $a_3$,
  • and finally from $C$ to $A$ as $b_1$, $b_2$, $b_3$.

Triangle triple product invariance

It is easy to show that the following products of lengths are equal to each other:
$a_1b_1c_1 = a_2b_2c_2 = a_3b_3c_3 \equiv v(P)$.

Using the barycentric coordinates of $P$ and the AM-GM inequality one can prove that $v(P)$ reaches maximum inside $ABC$ when $P$ is the centroid of $ABC$.

The equalities hold for the points outside of the triangle as well.

Alex C
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    FYI: In this ancient answer, I introduced an Extended Ceva's Theorem about the concurrence of (almost) any three lines with respect to a triangle and an equivalent relation among (combinations of) your lengths $a_i$, $b_i$, $c_i$. – Blue Mar 22 '25 at 05:24
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    I've seen the equation posted as a contest problem before (but can't easily find the source). It follows from similar triangles (which is essentially barycentric coordinates). – Calvin Lin Mar 22 '25 at 09:44

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