Here is a false proof. I will prove that a rectangle
inscribed in a square is a square. Return to my Mathematics pages
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© Copyright 2003, Jim Loy
Here is a false proof. I will prove that a rectangle
inscribed in a square is a square.
Proof: In the diagram, I have inscribed rectangle MNOP in square ABCD. Draw OR perpendicular to AB at R and PS perpendicular to BC at S. PS = OR. Draw MO and NP which intersect at Q. MO = NP. So triangle OMR is congruent to triangle PNS, and angle OMR = angle PNS. The angles BNQ and CNQ are supplementary. In the quadrilateral MBNQ, the angles QMB and BNQ are supplementary. So the angles B and MQN are also supplementary. But B is a right angle, and so angle MQN is also a right angle. And so the diagonals of rectangle MNOP are perpendicular. So quadrilateral MNOP is a square.
Flaw in the proof: Above, I said "In the
quadrilateral MBNQ, the angles QMB and BNQ are supplementary." Well, the
diagram may look like this (with triangle PNS a mirror image of the triangle
PNS in the above diagram), in which case, these angles are not supplementary.
We allowed ourselves to be led astray by the diagram, in the above proof.