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Insight into Solve...

Some insight requested...:

When Solving for the center of a circle that goes through two
specified points,

Solve[{(x1 - x0)^2 + (y1 - y0)^2 == r^2, (x2 - x0)^2 + (y2 - y0)^2 ==
r^2}, {x0, y0}]

the result gives expressions for x0 and y0 that are structurally very
different even though the symmetry of the problem says that they are,
in fact expressions that are ultimately very similar.

My question is what is the reason in the algorithm that Solve uses
that causes the initial results to structurally look so different.  It
appears that Solve is not aware of the symmetry in the problem.

Note that if instead of using x0 and y0 one used z0 and y0, then the
structural forms of the expressions are reversed suggesting that Solve
is taking variables alphabetically (no surprise here).

The problem with this sort of result from Solve is that one needs to
explicitly manipulate the resulting expressions to exploit the
symmetries in order for the final expressions to structurally/visually
exhibit those symmetries.  FullSimplify does not, starting from the
results of Solve, succeed in rendering the final expressions into the
desired form.

E.g, if temp is the result of the Solve command above,

 Assumptions -> {r >= 0, x1 \[Element] Reals, x2 \[Element] Reals,
   y1 \[Element] Reals, y2 \[Element] Reals}]

does not sufficiently reduce the expressions in to forms that
explicitly exhibit the symmetry transform into one another.

Is there another approach that comes to anyones' mind that will simply
yield the anticipated results?

Thanks in advance,


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