This JavaSketch demonstrates a theorem of Gérard Desargues, in two dimensions (or in three dimensions, if you choose to see this picture with 3D perspective). The point I is the intersection of the lines created by extending the orange side of each colored triangle. Points J and K are defined similarly. Drag the red points and observe the JavaSketch. Can you formulate what Desargues' Theorem is about? Can you prove the theorem? (Hint: it is easier to prove by starting in three dimensions.)
You can reset the JavaSketch to its original configuration at any time by hitting 'r' on your keyboard.