Regular pentagon erected internally on a side of a regular decagon
Regular pentagon ABKLM is erected internally on the side AB of a regular decagon ABCDEFGHIJ. Now vertex L of the pentagon is the midpoint of the decagon.
Since version 5.0.393.0 GeoGebra is able to check this property symbolically as well. Just click on the Relation tool and select segments b and then c, and finally choose "More...".
Activity
Check some other equalities concerning other segments in the decagon (or the pentagon) by joining some vertices and comparing the appearing segments with the Relation tool.
Theorems that are true on parts
Segments b and c are equal in fact just on parts. The reason behind this is that GeoGebra cannot distinguish between regular polygons and star-regular polygons. It considers all cases at the same time. Note that there is a star-regular pentagon, {5/2}, and there is also a star-regular decagon, namely {10/3}, as well. That is, all statements on the given construction here is considered on 2x2=4 different setups in the background!
Also, the segments BK and CD are parallel just on parts. A similar statement is true for segments KL and EF, too, for instance.