Desargues' Two-Triangle Theorem Part 1

Peg Fisher

We show that copolar triangles EFG and HIJ are coaxial through the following construction steps:

1. Construct lines AB, AC and AD. 2. Place E and H on line AC; F and I on line AB, and G and J on line AD (be sure the points E, F, G, H, I, or J are different from points B, C, and D). 3. Construct segments to form triangles EFG and HIJ. 4. Construct lines through each of the sides of both triangles. 5. Find the intersections (K, L, M) of the lines through EF and HI, FG, and IJ, and EG and HJ. 6. We checked the collinearity of the points K, L and M, using the input bar and entering "AreCollinear[K,L,M]" and the resulting boolean value c is true. (Note the styles of the various segments and lines to emphasize the goal of the construction.) Please feel free to move any of the points A, B, C, D, E, F, G, H, I, or J to test the validity of the construction. Thank you!