Consider any triangle ABC. There is
a point P in inside the triangle. Connecting vertices A, B, and
C to point P creates points D, E, and F.
Now consider lengths AF, BD, EC, FB,
DC, and EA. (AF)(BD)(EC)=(FB)(DC)(EA)!
The product of the ratios of these
compliments is always 1.
When P is moved outside the triangle,
the same property holds. To move Point P in the construction,
When P is brought outside of triangle
ABC, points D, E, and F can be moved outside of the line segments
that make up the triangle and onto the extended lines of the sides
of the triangle.
The ratio of the areas of triangles
ABC and DEF are affected by the position of P. As P moves closer
to the sides of triangle ABC, the ratio of the areas becomes greater.
The ratio is equal to 4 when P is on the centroid of the triangle,
dividing the triangle into 4 triangles of equal area. If P is
moved outside of the triangle, triangle DEF becomes larger than
triangle ABC and the ratio becomes smaller. To use the construction
yourself, CLICK HERE.
As P is moved to the centroid...
As P is moved outside triangle ABC...