This applet shows the construction for Desargues' theorem. Move the slider to see the construction progress. At step 0 we have three concurrent lines. You can move them using the diamond-shaped points. At step 1 we have the two triangles in perspective from a point. You can move the brown points along the lines to change the triangles. At step 2 we have matching edges of the triangles meeting in three points. At step 3 we have a line joining these three points -- Desargues' theorem says they will always be collinear.