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December 9, 2009 / guaild

Using TopSolid to exemplify theorems

Brianchon (http://en.wikipedia.org/wiki/Charles_Julien_Brianchon) proved that every theorem of projective geometry (http://en.wikipedia.org/wiki/Projective_geometry) has a “shadow” theorem which can be [deduced] by simply substituting nouns like “points” and “straight lines” and properties such as “to intersect” and “to lie on the same line”, or “inscribed” and “circumscribed”, or “side” and “diagonal” [Geometries of the Simulacra, Bernard Cache].

Brianchon’s theorem (http://www.cut-the-knot.org/Curriculum/Geometry/Brianchon.shtml) states that “opposite couples of points among the six vertices of the [hexagon] circumscribing an ellipse determine three pairs of side lines which intersect in one single point”. If we construct an ellipse that can be enlarged or shrunk on either of its two axes (in order to have a more general exemplification) and choose a set of six points on its periphery through which six tangent segments can be drawn, we can find the intersection point of each pair of segments and join opposing points. We find that, in fact, the three joining segments intersect at a single point (see fig. 1).

Screenshot of Brianchon's theorem exemplified in TopSolid

Fig. 1.

The “shadow” theorem of this one is Pascal’s theorem, which states that “opposite couples of points among the six vertices of a hexagon inscribed in an ellipse determine three pairs of diagonal lines which intersect in three points lying on the same line”.

Again, we can construct an ellipse (its axes are parametric and lie parallel to the xy axes for simplicity), on which we arbitrarily choose six points that describe the vertices of an irregular hexagon. Each pair of opposing sides are projected until they intersect (meaning that parallel opposing sides won’t work). The three points where these segments intersect lie on the same line (!!!) See fig. 2.

Pascal's theorem illustrated on TopSolid.

Fig. 2.

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