Triangle Investigation

By Karl Mealor

Problem (Parts A and B): 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)(ED) and (FB)(DC)(EA) for various triangles and various locations of P.

 

Click here for a GSP sketch to investigate:

 

Conjecture:

regardless of the triangle or position of P.

 

Click here for a proof of this conjecture:

Problem (Part C): 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?

 

Click here for a GSP sketch to investigate:

Conjecture #1: The ratio of triangles ABC and DEF equals 4 when P is located at the centroid. Click here for a justification of the conjecture:

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