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l"@D| @@z)Ɛ))@@D)9~Ԭ)99), 8@p94$,_9w&:Nrs'}mThe perpendicular bisectors of the sides of quadrilateral ABCD form a quadrilateral Q (unless ABCD is a rectangle), and the perpendicular bisectors of the sides of Q form a quadrilateral Q'. Show that Q' is similar to Q.
This problem was proposed by Josef Langr in the American Mathematical Monthly in 1953 [Problem E 1050, volume 60, p. 551], and brought to our attention by Branko Grnbaum in an article "Quadrangles, Pentagons, and Computers," in GEOMBINATORICS 3(1993) 4-9. No solution to the problem has ever been published.
The sketch above shows ABCD with Q in outline and Q' shaded. Lines have been constructed through what appear to be corresponding vertices of ABCD and Q'. The lines are concurrent at point P, and with Sketchpad, you can perform a dilation with center P and ratio
(edge of ABCD/corresponding edge of Q') that sends Q' onto ABCD. (To mark the ratio, first choose an edge of Q', then the corresponding edge of ABCD.) The ratio is negative when ABCD is convex, and positive when ABCD is nonconvex. Remember that a dilation with a negative ratio is the composite of a halfturn through P followed by the dilation through P with the corresponding positive ratio. By dragging any vertex of ABCD, the similarity of Q' to ABCD is maintained-- even in the self-intersecting cases. So Sketchpad "proves" the theorem-- we still await a conventional proof.
Grnbaum challenges us to find a full proof of Langr's problem and determine the ratio of the dilation which somehow depends on the shape of ABCD. Other problems to be solved:
(a) determine for what quadrilaterals ABCD the quadrilateral Q is similar to ABCD;
(b) investigate what happens with pentagons and with hexagons.
Do we call Langr's problem a theorem? Is the strong evidence provided by Sketchpad enough to declare that there is a theorem, even though we don't have a conventional proof?
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