desargues’ theorem in color

March 27, 2015 in desargues

the diagram is, essentially, traced from
bruce e.~meserve’s _fundamental_concepts_-
_of_geometry_ (dover reprint of 1983;
originally addison-wesley 1959). but
i added in the colors. with which, one
has as follows.

there’s a red line, a blue line, and a yellow line.
to start with. all sharing a point.
then two “triangles” are constructed:
each of these is to have a red, a blue, and a yellow
“vertex”.
(on the diagram, the “red”, “blue”, and “yellow”
vertices of one of the triangles is actually black;
this helps [maybe… it helps *me*] in determining
which vertices belong to which triangles.)

next we can think of the *edges* of these two triangles
as “blending two primary colors”; for instance we can
think of the red-and-yellow edge of either triangle as
determining an *orange* line… and go on to construct
an “orange vertex” at the intersection of these lines.

likewise, the yellow-and-blue edges of the two triangles
will determine a *green* point (at the intersection of
two green lines) and the blue-and-red edges will determine
a *purple* point.

desargues’ two-triangle theorem then says that the
orange, green, and purple points will “line up”.

proof: meserve (or any of *many* other sources)

footnote: alas, one has *exceptional* cases
in “euclidean two-space”, i.e., the ordinary
(two-dimensional) *plane* of high-school
geometry. one remedies this by working
instead in a (so-called) *projective* plane
(in such a plane, there are no “parallel” lines;
nevertheless, much of “ordinary” plane geometry
becomes *easier* in projective geometry [example:
this theorem]).

putting color names on things is a common trick in math,
but if anybody else is using blends-at-intersections in
anything resembling this way (elementary projective
geometry, i suppose i mean), i don’t know about it.
(priority claim; if you use this amazingly good idea,
remember where you got it. please.)

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