A necessary not-sufficient condition
- 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.