In geometry, the crossbar theorem states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of…

3 Instead of considering two arbitrary chords intersecting, consider one arbitrary chord with endpoints S and T on a circle with center C and one arbitrary point X on the chord ST.

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