Given a triangle ABC and two midpoints D and E. Point D is interesting because

two parallel lines appear as a consequence

of the Midline Theorem. Now we create the third midpoint F. Point F is interesting since it

produces two pairs of parallel lines. Varignon’s Theorem claims that midpoints of

an arbitrary quadrilateral designate a parallelogram. We are interested if point H is related to

some non-trivial relationships. Yes. There are two sets of parallel lines.

Both sets are colored with different colors. Now let us create the midpoint

of segments AH and AE, respectively. The discovery about the interesting facts of point I

already starts when typing the command. The resulting set of interesting facts will be

very quickly displayed. Note that the various colors designate

different directions of parallel lines. Euler’s line is a well known object in planar geometry.

It connects the orthocenter F, centroid G

and circumcenter H. The Discover command finds not only the collinearity of F, G and H,

but also the concyclicity of E, H, D and C. In fact the two pairs of parallelism should not be reported

since these lines are parallel to each other by definition. But this question may be handled a bit flexibly. The nine-point circle of a triangle is designated by

the feet of the altitudes (D, E, F), the midpoints (H, I, J)

and the midpoints of the orthocenter G and the vertices (K, L, M). In fact not only concyclicity

but also several parallelisms occur. The Relation Tool informs the user on the fact of parallelism. After a numerical check a strict proof can be performed. The Relation command can inform the user on the concyclicity as well. At the moment most lines are colored with the same color.

Maybe this is just a bug and should be fixed. Also, some lines are unfortunately duplicated. Now, after creating a triangle,

a lot of random midpoints will be generated. We will test the robustness of the Discover command on many objects. Usually it takes just a few seconds to get the result. A second Discover command will use the previously discovered facts.

But each computation may take a bit longer when new points are considered. The Discover command works also in the web version of GeoGebra,

that is, in GeoGebra Classic 6. A regular hexagon will be created. Now we are checking if the point B is interesting. The Discover command appears in a context menu of each point. Due to an internal problem first an error message was shown. But then both the concyclicity of the vertices and

several sets of parallel lines are detected correctly. However, the Discover command performs a bit slow in the web version.