LW

Above is an animated version of Figure 2 from the paper, showing a geodesic in triangle space. The geodesic starts at the equilateral triangle shown, and the three curved paths show the tracks of the three vertices.

chium!

(via 238970a8e3c54b99f79c20f4dfb65386daea0397_m.gif (GIF Image, 480 × 480 pixels))

(via https://www.youtube.com/watch?v=-Sn2BdUGDbg)

Simson line theorem. The three blue points always lie on a straight line. The blue points are the closest points to the moving red point on the lines. In other words the blue points are the projections of the moving red point to the lines.     

The count grows simply as 8n - 4. How would you show that?

(Source: Adam Plouff)

© Geometrymatters,2014

This is actually a very special case of Ptolemy’s theorem. The theorem gives a connection between the sides and the diagonals of a cyclic quadrilateral. In this case the length of the dashed lines is equal so the theorem can be simplified to the statement above.