Harmonic Relation of 4 points on a line.

Proposition: H(RT,SU) is equivalent to H(SU,RT) (adapted from Meserve & Izzo). I.e., if H(RT,SU) then H(SU,RT).
Recall: Four points on a line l are harmonically related if the line is determined by a pair of points from the intersection of lines in a complete quadrangle and the intersection of that line with the other two sides of the complete quandrangle.
[In the figure: The line XZ would determined two other points, XZ#AD=R and XZ#BC=S, so that the points XRZS are harmonically related.] This is denoted H(XZ,RS).

Four points on a line that are  harmonically related: It has already been demonstrated that  if H(AB,CD) then H(BA,CD), and conversely if H(BA,CD) then H(AB,CD). This is the meaning of saying "H(AB,CD) is equivalent to H(BA,CD)". Similarly H(AB,CD) is equiv. to H(AB,DC) and H(BA,DC).

Notice the relation between the "double points" in the figure and the "single points".

Proof of Proposition:(adapted from Meserve & Izzo).
Consider the new quadrangle determined on the figure by the four points P3, P4, W and V.
[To complete the proof we need only show that WV meets SU at the point T.]
Notice in the figure that Triangle WP1P2 is perspectively related by the line SU to triangle VP3P4.
Thus by converse of Desargues' Theorem we have that the triangles are perspectively related by a point.
But this point must be T, so the line WV passes through the point T, completing the demonstration that H(SU,RT).

Sorry, this page requires a Java-compatible web browser.