Re: Insight into Solve...

*To*: mathgroup at smc.vnet.net*Subject*: [mg89838] Re: [mg89815] Insight into Solve...*From*: bsyehuda at gmail.com*Date*: Sun, 22 Jun 2008 03:23:43 -0400 (EDT)*References*: <200806210931.FAA15784@smc.vnet.net>

Hi David, not automatic as I would like, but after the first FullSimplify you can use a replacement rule ... /. (x1-x2)^2+(y1-y2)^2->d^2, and the Fullsimplify again with all the previous assumptions with the addition 0<=d<=2r you will notice that symmetry is more apparent I hope this helps p.s. Appart on the last expression (after second FullSimplify) may be even clearer yehuda On Sat, Jun 21, 2008 at 12:31 PM, David Reiss <dbreiss at gmail.com> wrote: > > > 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, > > FullSimplify[circlePoints, > 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, > > David > >

**References**:**Insight into Solve...***From:*David Reiss <dbreiss@gmail.com>