A generalization of Simson-Wallace
- Draw a triangle ABC and create point D somewhere.
- Reflect D about all sides of the triangle.
- All the mirror images of D collinear? Usually no.
- Ask hints from GeoGebra where to put D in order to have the mirror images collinear. Type
LocusEquation[AreCollinear[D',D'₁,D'₂],D]. - Does this hint remind you of the Simson-Wallace theorem?
- Can you find a generalization of Simson-Wallace by using your discovery? Yes, since both the mirror images and just the projections are collinear if D is properly selected, it seems valid that any point on the line joining the mirror images and just the projections will result in something similar.
- Can you find a special case of this conjecture to check the statement? Yes, for example the projections can be mirrored on the mirror images, respectively. This can also be checked by using GeoGebra Automated Reasoning tools (namely, the Relation tool).