The Final set of explorations are found here in this section. In the first one, a point is taken at random inside a triangle and the lines are drawn from each vertex to the opposite side and passing through this random point. An interesting relationship is noticed between the ratios of the segments thus created.

In the second, a random point is selected inside a given triangle. Then, lines are drawn from the vertices of the triangle, passing through point P till they meet the sides of the triangle. A new triangle is constructed using these points as the vertices. The relationship between the area of this new triangle and the parent triangle is explored here.