Harmonic Relation of 4 points on a line.
Proposition: H(RT,SU) is equivalent to
H(SU,RT) (adapted from Meserve & Izzo).
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,
so that the points XRZS are harmonically related.] This is denoted
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
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).