ANALYSIS:
Let a, b, c, and d denote lines that are constructed
through points A, B, C, and D respectively such that
c||a and a⊥b and b||d.
The intersections of these lines
P = d∩a
Q = a∩b
R = b∩c
S = c∩d
are the vertices of the rectangle PQRS.
Clearly, the decription for the construction of only
one of these four lines, say d, is needed as the
other three follow per definition.
CONSTRUCTION:
If BD⊥AC then
if |BD|=|AC| then
Construct d||AC.
else
Construction not possible.
else
Construct line g through B⊥AC. The circle
with center B and radius |AC| intersects line g
at two points. Label one of the points D' such
that DD' and AC are not parallel. Construct line
d through D'.
PROOF:
If BD⊥AC then
if |BD|=|AC| then
clearly, PQRS is a square since
|PS| = |AC| = |BD| = |PQ|.
else
clearly, the construction when |BD| = |AC| will
not work. If d is constructed perpendicular to
AC, then P = Q = R = S = AC∩BD. But, what is
the meaning of point X lies on line YZ if Y = Z?
Assume line d is any line through point D (that
is not parallel or perpendicular to AC) which
forms a square PQRS. Construct a line through
S||AC intersecting line a at A' and a line
through S⊥AC intersecting line b at B'.
ΔSPA' ~ ΔSRB' since corresponding sides are
perpendicular.
|SB'| = |BD| ≠ |AC| = |SA'| ⇒ |SP| ≠ |SR|.
This contradicts the assumption that
PQRS is a square.
else
Construct a line through S||AC intersecting line
a at A' and a line through S⊥AC intersecting line
b at B'.
ΔSPA' ~ ΔSRB' since corresponding sides are
perpendicular.
|SB'| = |BD'| = |AC| = |SA'|. ⇒ |SP| = |SR|.
Therefore, PQRS is a square.
QED
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