Desargues' Theorem/Converse
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Theorem
Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles lying in the same or different planes.
Let:
- $BC$ meet $B'C'$ in $L$
- $CA$ meet $C'A'$ in $M$
- $AB$ meet $A'B'$ in $N$
where $L, M, N$ are collinear.
Then the lines $AA'$, $BB'$ and $CC'$ intersect in the point $O$.
Proof
Let $L$, $M$ and $N$ be collinear by hypothesis.
Then $\triangle BB'N$ and $\triangle CC'M$ are perspective with center $L$ ($L = BC \cap B'C' \cap MN$)
From Desargues' Theorem:
- $O = BB' \cap CC'$
- $A = BN \cap CM$
- $A' = C'M \cap B'N$
are collinear.
Thus:
- $AA' \cap BB' \cap CC' = O$
Hence $\triangle ABC$ and $\triangle A'B'C'$ are perspective with center $O$.
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$\blacksquare$
Also see
Source of Name
This entry was named for Girard Desargues.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Desargues' theorem
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Desargues' theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Desargues' theorem