Desargues Theorem

The green and yellow triangles are projections of each other from the point where the lines A1A2, B1B2 and C1C2 meet.

If the corresponding edges of the triangles are extended to meet (e.g. A1B1 and A2B2 extended till they meet at AB), the three meeting points lie on a line AB-BC-CA.

This is called Desargues theorem, and is one of the fundamental theorems of projective geometry.

You can move the point of projection, the projection lines, and the vertices of the red and green triangles. The points AB, BC, CA will move, but always remain collinear.

Try making the triangles overlap, or dragging one of the projection lines to change the order of lines.

It is sometimes easier to see why this is so if you look at the diagram as a representation of a 3D pyramid. The pyramid is sliced by two planes, giving the triangs A1B1C1 and A2B2C2. The two planes meet along a line, which is AB-BC-CA.

 

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