# A necessary not-sufficient condition

- Author:
- Zoltán Kovács

This applet shows GeoGebra's animation possibilities by using fast real-time Gröbner basis computations.
After constructing a triangle, the command

`LocusEquation[a+b==5,C]`

was used. For technical reasons the point *A*is not attached to line*f*, but the slider*d*conducts the animation by explicitly setting the*x*-coordinate of*A*. Note that the geometrical solution should be an ellipse with foci*A*and*B*in such cases when*c*<5, because the triangle inequality ensures*a*+*b*>c, so there is room for possible solutions. Otherwise, if*c*=5, we obtain*a*+*b*=*c*, that is*C*must lie on the segment*AB*; actually in this case the line*AB*will be shown in GeoGebra because no further restrictions can be achieved by using the Gröbner basis approach. (In other words: the line is algebraically a necessary, but geometrically not-sufficient condition.) Finally, if*c*>5, by using the triangle inequality again, we should not find any solutions, but GeoGebra delivers a hyperbola with foci*A*and*B*: the reason here is that there is no way in the Gröbner basis approach to exclude the hyperbola which is algebraically a necessary condition.