**Lets choose a triangle of no particular
type and a random point within the triangle and draw lines from
each vertex point through the chosen point to an intersection
with the opposite side.**

**Using the measure tool in GSP we arrive
at the following measurements:**

**So we see that the two products are equal.
We may make the conjecture that this will always be the case.
For a working sketch in gsp to check the numerical validity of
this conjecture click here.**

**The calculation of the area of the triangles
ABC and DEF are given here:**

**For a working gsp sketch which calculates
the ratio of these areas for any point p in the interior of ABC
click here.**

**This ratio will always be equal to or greater
than 4. We would conjecture that the ratio would be equal to 4
when triangle DEF connects the midpoints of the sides of triangle
ABC. A welll known theorem from geometry states that a segment
connecting the midpoint of the sides is parallel to the thhird
side and half as long.Lets check this out.**