One of the fundamental theorems of projective geometry is
Desargues Theorem, which concerns the relationship of two triangles.

Open up Sketchpad
Construct a triangle and label the vertices A, B and C.
Create a point off the triangle and label it P.
Connect P and B, P and C, and P and A with segments
Choose points on these segments and label them as A', B' and C'
respectively.
Connect the segments to form the triangle A'B'C'

Create the lines AB and A'B' and find and label their intersection.
You may have to move some of your initial points in order to fit the entire diagram on the screen,
including the intersection points.
Similarly, create the intersection between AC and A'C'
Finally, create the intersection between BC and B'C'

- Do the three intersection points lie on a line?
- Desargues' Theorem (roughly) states that 2 (projective) triangles are
perspective with respect to a point (P above)
if and only if they are perspective with respect to a line (of
the intersections).
Does the theorem hold as you drag points around
in your figure?
- What happens if you drag B' to make A'B' parallel to AB?
- Does Desargues' Theorem hold in hyperbolic geometry or in
spherical geometry? Recall that Sketchpad like
programs are found in
Help/Sample Sketchpad/Advanced Topics (for hyperbolic) and
Spherical Easel.