# computational geometry proofs; systems of equations

Robert Lewis lewis at bway.net
Mon Dec 6 15:03:23 PST 2004

```Hello,

I am interested in establishing geometrical results (not necessarily
theorems) by computer algebra, specifically via the solution of
multivariate polynomial equations.  I have used the Dixon resultant
technique to do so, for example,

Lewis, Robert H. and Stephen Bridgett, "Conic Tangency Equations and
Apollonius Problems in Biochemistry and Pharmacology," Mathematics and
www.bway.net/~lewis/lewbrid.pdf

Some computational geometric proofs using a different technique are on

I wonder if anyone has tried (with such techniques) before to prove
Ptolemy's theorem: For a quadalateral inscribed in a circle, let d1 and
d2 be the (lengths of the) diagonals, d3 and d4 one pair of opposite
sides, d5 and d6 the other. Then d1*d2 = d3*d4 + d5*d6.

I have done so by getting 9 equations in the 6 di and 8 other variables
that represent the sine and cosine of various angles. The resultant is
indeed d1*d2 - d3*d4 - d5*d6. It takes about 40 minutes cpu time to do
it.

I have also proved the converse: assuming d1*d2 - d3*d4 - d5*d6 = 0,
the quadralateral is on a circle. This actually takes far less time,

Also, is there any information on similar three dimensional results?

Regards,

Robert H. Lewis
Fordham University
http://www.bway.net/~lewis/

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