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In his book "Plane geometry and its groups", H. Guggenheimer proves the Stewart's formula:

If $A$, $B$, and $C$ are collinear, then for any point $Ρ$ in the plane $$ PA^2 BC + PB^2 CA + PC^2 AB + AB . BC . CA = 0. $$

and states that Stewart's formula [...], in principle, solves all computational problems in plane geometry.

What does he mean ? How do we solve all computational problems in plane geometry using that formula?

I understand that, for example, formulas given by the Gram determinant of the inner product of vectors are fundamental in the sense that two sets of vectors can be deduced from each other using an isometry if and only if these Gram determinants are the same, and this statement can be interpreted in terms of Invariant Theory. But I fail to find such an interpretation to Stewart's formula. What is the justification to his bold statement?

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    $\begingroup$ It possibly just means that many problems reduce to finding the fifth member of the set $\{PA,PB,PC,AB,AC\}$ given that other four are known. Another interpretation is that Stewart's formula is universal in a philosophical sense: it is an equivalent substitute to Cartesian coordinates, which are universal in the usual sense. $\endgroup$
    – Fedor Petrov
    Commented Jun 24, 2020 at 14:32
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    $\begingroup$ can't you say the same about Pythagoras? (since Pythagoras --> law of cosines --> Stewart) $\endgroup$
    – Carlo Beenakker
    Commented Jun 24, 2020 at 17:02
  • $\begingroup$ mathoverflow.net/questions/234184/… $\endgroup$
    – Đào Thanh Oai
    Commented Jun 26, 2020 at 2:05

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