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:
- Construct segment AB
- Construct
perpendiculars to
segment
AB at the endpoints A and B.
- Attach a point C to one
of
the perpendiculars.
(To attach a point, just create the point on top of the line)
- Construct a segment
from
point B to
point C. Note that this segment will lie on top of the perpendicular
below
it.
- 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.
- Find the intersection
of
the circle
and the perpendicular at A. Call this point D.
- 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?