Given a triangle ABC.
Construct triangle DEF as follows:
Extend a ray from A through B and locate D such that AB = BD
Extend a ray from B through C and locate E such that BC = CE
Extend a ray from C through A and locate F such that CA = AF
What can be said about these two triangles and any relationships between them?
Click HERE for a GSP 4.0 file to explore this construction.
Show that triangle ABC is not similar to triangle
DEF.
Do triangle ABC and triangle DEF have the same
centroid?
Construct
points G, H, and I on DE, EF, and FD respectively by extending
AC to intersect with DE, BA to intersect with EF, and CB to intersect
FD.
Show that
Show that G, H, and I are trisection
points of the sides DE, EF, and FD respectively.
Construct
the other trisection points on each side label them J,K, L.
Construct FJ, DK, and EL. Make their intersection M, N, and O.
What can be said about triangles ABC and MNO?
What fraction of the area of Triangle ABC is the GREEN irregular hexagon formed by the overlap of triangles ABC and MNO?
Click HERE
for a GSP 4.03 file with this construction.
Extensions
1. Explore this problem with ABC constructed as an EQUILATERAL triangle.
2. Explore with Triangle ABC a right triangle.
3. Develop the construction with the vertices of triangle DEF only one-half the extension of the sides of ABC
