GEOMETRIC SITUATION #1:

We draw a triangle and the center of the circle passing through the three vertices of the triangle (the triangle's circumcenter). Profiting from Cabri-Geometric features, we drag the vertices, observing that in some instances the center of the circle lies on a triangle's side. Measuring, in such cases, the opposite angle, we conclude that it is a right angle. The converse statement (on a right triangle the cicumcenter always lies on the hypothenuse) can be likewise verified.

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dibujo1

We will show, on this simple case, how a computer algebra system is able to automatically "discover" the same result. First of all we must establish a (wrong) conjecture, just involving the given construction, such as: on every triangle the circumcenter lies on one side. Therefore we take as hypotheses the given construction (the given vertices, the center of the circle). As thesis, we state that such center lies on a side. The system will determine that the thesis is generally false; and that it is true if and only if we have a right triangle.

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