Saccheri Quadrilateral
Girolamo Saccheri
(1677-1733) was an Italian mathematician who for many years tried to prove
the fifth Euclidean axiom from the other four. He based his proof
attempts on the idea of negation. If he assumed Euclid 1-4 as axioms and
assumed the opposite of Euclid 5 and then reach a contradiction to something
known to be true, then the opposite of Euclid 5 could not hold simultaneously
with the first 4 axioms, and thus Euclid 5 must be true as a consequence
of the other four axioms.
One of
his proof attempts involved the construction of a quadrilateral ABCD in
which the sides AD and BC are assumed equal and there are right angles
at points A and B. To construct this figure:
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Construct segment AB
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Construct perpendiculars to segment
AB at the endpoints A and B.
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Attach a point C to one of the perpendiculars.
(To attach a point, just create the point on top of the line)
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Construct a segment from point B to
point C. Note that this segment will lie on top of the perpendicular below
it.
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Select point A and then the segment
BC (you may have to click on the segment twice to select just the
segment) Then click on the circle construction tool in the Construction
section of the Tool Panel. This will create a circle at A of radius BC.
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Find the intersection of the circle
and the perpendicular at A. Call this point D.
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Attach D to C.
The
Saccheri Quadrilateral has some interesting properties. One of these is
shown. If we measure the two upper angles, these appear to be always congruent,
but never 90 degrees! Can you prove this?