Consider triangle ABC. A point P was selected randomly inside the triangle. segments were drawn from P to each of the triangles vertices and then extended to intersect the sides of the triangle.

Let's explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.

First I will explore for the triangle shown above.

Notice that the products are equal.

Now I will move the location of P.

The product remains equal!

If I make triangle ABC an obtuse triangle, the same relationship holds.

Conjecture: Can the results be generalized so that P can be OUTSIDE the triangle?


here I just extended the segments into dashed lines.

Notice that when P is moved to the exterior of the triangle, the other lines disappear.

Here, I recreated the triangle constructing it with lines instead of segments.

Now, when I move the point P outside of the triangle, I can see that the same relationship holds.

Click HERE to move point P around.

Next, I will show that when P is inside triangle ABC, the ratio of the areas of triangle ABC and triangle DEF is always greater then or equal to 4.

Notice that in this particular case, it is greater than 4.

Click HERE to look at all possibilities.

We can see from the picture below that the ratios of the areas will equal exactly four when the medial triangle is constructed.