We use Gröbner basis techniques to prove that the three medians of a triangle are concurrent. The common intersection point is called the centroid of the triangle.
We may choose the coordinates of the vertices of the triangle to be as follows: \[A: (0,0) \quad B: (b,0) \quad C: (x,y)\] where $b,x,y$ are indeterminate. We denote the intersection of the median through $A$ and the median through $C$ by coordinates $(a,c)$, again with $a,c$ indeterminate.
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The above is a shorthand notation for creating a polynomial ring with total degree ordering.
Three points $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ in the plane are collinear if and only if \[\left|\begin{array}{ccc}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{array}\right| = 0.\]
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The following polynomials are expected to be zero. They express the fact that we know three points on the median through A and three points on the median through C.
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And the following polynomial expresses what we want to prove
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Consider the ideal generated by the two assumptions.
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The conclusion should be part of this ideal, i.e., it should reduce to 0 through $G$
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QED.
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