### desargues’ theorem in color

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

determiing 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.)

March 27, 2015 at 3:28 pm

i’ve been doing this for a *while*:

https://vlorbik.wordpress.com/2011/07/17/desargues-theorem-for-poets/

March 27, 2015 at 4:10 pm

http://commons.wikimedia.org/wiki/User:David_Eppstein/Gallery#Configurations_and_point-line_incidences

May 2, 2015 at 6:09 pm

https://www.librarything.com/work/357686/book/70016092

meserve