Prove Desargues’ Theorem for the case where AC is parallel with A′C′ on the extended Euclidean...

Prove Desargues’ Theorem for the case where AC is parallel with A′C′ on the extended Euclidean plane.

Expert Solution

Triangles ABC and A′B′C′ lie on the same Euclidean plane and their corre-
sponding vertices (point A corresponding to point A′, B to B′, and C to C′) are
connected by lines that all meet at a single point (O). The two triangles are said
to be ‘in perspective from O’. Desargues’ theorem concerns pairs of correspond-
ing sides of the two triangles, where side CA corresponds to C′A′, for example.
Line CA may intersect line C′A′ (remembering that each of these lines extends
infinitely beyond the segment forming a side of one of the two triangles inperspective); unless CA and C′A′ are parallel, they will intersect somewhere on
the plane. In the figure, M is their point of intersection. Likewise, N lies at the in-
tersection of AB and A′B′, and L lies at the intersection of CB and C′B′. The the-
orem says that these three points of intersection, if they exist, are collinear.

milcah answered 1 year ago

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