by Jan White
Consider any triangle ABC. Select a point P inside
the triangle and draw lines AP, BP, and CP extended to their intersections
with the opposite sides in points D, E, and F respectively. Explore
(AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations
As can be seen the ratio is 1. The ratio of similar segments of similar triangles will always be one. For an animation when point p is inside triangle ABC click here.
Show that when P is inside triangle ABC, the ratio of the areas of triangle ABC and triangle DEF is always greater than or equal to 4. When is it equal to 4?
As long as point P stays within the triangle ABC then the ratios of their areas will always be equal to or greater then 4. Click here for an animation. The ratios of their areas will be equal to four when points F, E and D are located at the medians of AB, AC, and AC as this is where triangle FED will have its greatest area. For an animation of this click here.