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This is a tricky one to describe.
Take a sheet of blank paper. To the left, draw a single point, and label
it O. Now, draw three rays that emanate from this origin point, extending
to the right. One of the rays can be horizontal, but try to draw them
such the angle between the top ray and the middle ray is noticeably
different from the angle between the middle and the lower ray. Label
these rays, from top to bottom, a, b, and c, respectively.
Now, pick three points, one from each of the three rays, such that
when you form a triangle by connecting these points, your triangle is
clearly visible (in other words, not too flat). Connect these points,
and call these triangle vertices a1, b1, and c1, corresponding to the
ray on which they lie.
Next, repeat this procedure, but choose three more points, this time
farther to the right of the first group. It would be helpful if the
two triangles do not appear to be similar, but this is not essential.
Call these vertices a2, b2, and c2.
If you have completed the construction properly, you should have a
picture of two triangles, with lines Oa1a2, Ob1b2, and Oc1c2 all being
collinear. These two triangles, one projecting through the other to
a single point O, are said to be in perspective.
Now, there is a single line that passes through points a1 and b1. Use
dashes to sketch this line. Likewise, there is a line that passes through
points a2 and b2. Sketch this line, and note the point of intersection
of the two dashed lines. These two dashed lines are referred to as corresponding
lines. There are two other pairs of corresponding lines, corresponding
to a with c, and b with c.
Desargue's Two Triangle theorem says this: If two triangles are in
perspective, then the points of intersection of their corresponding
line pairs fall on a straight line.
It is important to mention that while the construction just described
takes place on a plane (the piece of paper), the theorem actually extends
to triangles not in the same plane, provided they are in perspective.
The theorem was proposed and
proven by Girard DeSargues, around 1630. While most mathematicians of
the day pursued their studies out of a sense of the pursuit of knowledge,
or because the beauty and aesthetics of the discipline enraptured them,
Desargues' reasons for furthering projective geometry were more practical
in nature. He worked as a geometer/engineer/architect, and was dismayed
by the dismal state of technical drawing. Because of this concern, he
worked in projective geometry towards a development of perspective theory,
specifically so that his geometric ideas might make the drawing around
which his work revolved more precise and reliable.
This theorem, with others
provided the basis for perspective theory.
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