Desargues' theorem


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Desargues' theorem

[dā′zärgz ‚thir·əm]
(mathematics)
If the three lines passing through corresponding vertices of two triangles are concurrent, then the intersections of the three pairs of corresponding sides lie on a straight line, and conversely.
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Mathematics: ABEGG'S RULE, ABEL'S THEOREM, ARCHIMEDES' PROBLEM, BERNOULLI'S THEOREM, DE MOIVRE'S THEOREM, DE MORGAN'S THEOREM, DESARGUES' THEOREM, DESCARTES' RULE OF SIGNS, EUCLID'S ALGORITHM, EULER'S EQUATION/FORMULA, FERMAT'S PRINCIPLE, FOURIER'S THEOREM, GAUSS'S THEOREM, GOLDBACH'S CONJECTURE, HUDDE'S RULES, LAPLACE'S EQUATIONS, NEWTON'S METHOD/PARALLELOGRAM, PASCAL'S LAW/TRIANGLE, RIEMANN'S HYPOTHESIS
Desargues' Theorem for perspective triangles: The connecting lines of the couples of corresponding vertices of two triangles ABC and ABC intersect at a point S if and only if the intersection points of the couples of corresponding sides P=BC[intersection]B C', Q=AC[intersection]A C', R=AB[intersection]A B lie at a line s (Figures 16.
Definition 5: Two triangles that satisfy the conditions of Desargues' Theorem for perspective triangles are called perspective.
Let us mention that Desargues' Theorem about perspective triangles does not restrict us to choose the vertices of the two perspective triangles to lie on two lines.
Solution: The point S is a perspective center of the pair of triangles AB'C and ABC' and according to Desargues' Theorem the intersection points P, Q, T of the couples of corresponding sides are collinear.
According to Desargues' Theorem, applied to [DELTA]MSD and [DELTA]RNB, the lines MR, SN, DB are concurrent if and only if the points F=MS[intersection]RN, A=MD[intersection]RB, C=SD[intersection]NB are collinear.
According to Desargues' Theorem, applied to [DELTA]MRC and [DELTA]SNA, the lines MS, RN, CA are concurrent if and only if the points E=MR[intersection]SN, T=RC[intersection]NA, G=MC[intersection]SA are collinear.
We do not need to evoke Goedel's theorem to prove this; Desargues' theorem about triangles in perspective will do.