| Article 6977 of sci.math
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From: tycchow@phoenix.Princeton.EDU (Timothy Yi-chung Chow)
Newsgroups: sci.math
Subject: Re: Quadrilateral Problem
Message-ID: <10494@phoenix.Princeton.EDU>
Date: 19 Sep 89 13:42:48 GMT
References: <846@acf5.NYU.EDU> <88.UUL1.2#239@valley.UUCP>
Reply-To: tycchow@phoenix.Princeton.EDU (Timothy Yi-chung Chow)
Organization: Princeton University, NJ
Lines: 32
In article <88.UUL1.2#239@valley.UUCP> stan@valley.UUCP (Stanley L. Kameny) writes:
>> On the sides of a convex quadrilateral are erected externally directed
>> squares. Show that the centers of these squares form a square. What if the
>> squares are directed internally?
>
>For one thing, it is easy to find counterexamples. Neither case works
>if the quadrilateral has two very small adjacent sides, and two long
>sides.
>
>The problems must be stated incorrectly!
O.K., I guess it's time for me to clear up the situation. I've already
e-mailed a solution to the original poster but it looks like I should
post an article.
The correct statement of the problem is: Show that if you join the
centers of opposite squares, the two resulting line segments are equal
in length and perpendicular. (This is actually true even if the
quadrilateral is concave or "degenerate.")
There is also a related problem: Take the midpoints of the two line
segments mentioned above, and take the midpoints of the diagonals of the
original quadrilateral. Show that these four points form a square.
I posted this problem to rec.puzzles a couple of months ago but as far
as I can recall, no correct solutions were posted.
HINT FOLLOWS:
Embed the quadrilateral in the complex plane and note that
multiplication by i is equivalent to a 90 degree rotation.
|
| >> On the sides of a convex quadrilateral are erected externally directed
>> squares. Show that the centers of these squares form a square. What if the
>> squares are directed internally?
While this statement is unprovable, the equivalent statement for triangles
is provable; i.e. that for any triangle, the centers of externally directed
equilateral triangles erected on the three sides are the vertices of an
equilateral triangle.
LAAEFTR (no, not laughter, but Left As An Exercise For The Reader).
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| Along similar lines to .2, if you intersect the appropriate trisectors
of the angles of an arbitrary triangle, pairwise, then the points are
the vertices of a little equilateral triangle.
- Jim
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