Proof - The Line Connecting the Midpoints of a Saccheri Quadralateral is Perpendicular to Both of the Lines.
The Saccheri Quadralateral
The Saccheri Quadralateral was created by Girolamo Saccheri (1667-1733), a
professor at the University of Pavia, in an attempt to prove Euclid's fifth
postulate by contradiction. While it did not fulfill its original purpose, the
Saccheri Quadralteral has become an important part of Hyperbolic Geometry.
The Saccheri Quadralateral is constructed by taking an arbitrary line (AB in
the diagram), and constructing two lines of equal length (AD and BC) that are
perpendicular to the original line (AB). The points D and C are then connected
with a line.
To prove that the line FE is perpendicular to both BC and AD, we must first prove
that the summit angles (angle ABC and angle DCF) are congruent.
Proof that the Summit Angles of a Saccheri Quadralateral are Congruent
- To show: That angle ABC is congruent to angle BCD.
- It is given that angle BAD and angle CDE are congruent and both measure 90 degrees.
- Since AB is congruent to DC by construction, and AD is congruent to AD by same line,
we can conclude that triangle ABD is congruent to triangle DCA by SAS.
- Since corresponding parts of congruent triangles are congruent (CPCTC), we can
conclude that BD is congruent to CA.
- Since CD is congruent to CD by same segment, we can conclude that triangle BCD is
congruent with triangle CBA by SSS.
- We can now conclude by CPCTC that angle ABC is congruent to BCD. QED.
Proof that the Line Connecting the Midpoints of the Base and Summit of a
Saccheri Quadralateral is Parallel to Both.
- To show: That EF is perpendicular to both BC and AD.
- BF is congruent with CF by construction, and BA is congruent to CD by construction,
and angle ADC is congruent with and BCD as previously proved. We can then conclude
that triangle ABC is congruent with triangle DCF by SAS.
- AF is congruent with DF by CPCTC. Also, AE is congruent to DE by construction, and EF
is congruent to EF by same segment. We can then conclude that AFE is congruent to
DFE by SSS.
- Angle AEF is congruent to angle DEF by CPCTC. Since they are supplementary angles, they
must be right angles. Therefore EF is perpendicular to AB.
- By a similar argument, triangle BEF is congruent to triangle CEF, and EF is perpendicular
to BC. QED.