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Re: Plausibility Arguments
Posted:
Mar 13, 1997 4:33 PM
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On Thu, 13 Mar 1997, Ted Alper wrote:
> Draw two intersecting circles, of somewhat different radius -- call
> their centers A and B (call the intersection points C and D). From C
> draw the diameters for both circles (CE and CF, where CE goes through
> A and CF goes through B). Now draw line EF, and label the points where
> it intersects circle A X and label the point where it intersects
> circle B Y. CXE is a right angle, since CE is a diameter of circle A;
> CYF is a right angle, since CF is a dimater of circle B; X,Y,E, and F
> are colinear; therefore angles CXY and CYX are right angles... so
> triangle CXY has two right angles!
Thank you, Ted, for this example, I did not know it.
There are several examples which I like. One of them is
a proof that an arbitrary triangle ABC is isosceles.
Denote M the center of AC and m the line, which goes
through M and is perpendicular to AC. Denote b the bissector
of the angle B. Denote N the point of intersection of m and b.
I assume that N is outside ABC (which in fact is true), but the
other case (if we assume that it is possible) can be considered
analogously. Drop perpendiculars NK and NL from N to the lines
BA and BC respectively. It is easy to prove that triangles BNK
and BNL are congruent, whence BK = BL. Also it is easy to prove
that the triangles NKA and NLC are congruent, whence KA = LC.
Now subrtact these equalities and you get BA = BC!
This example (however, with a confused picture) was included
into the useful book `Mathematical Circles' which is published
by AMS and a review of which, written by me, will soon appear
in the American Mathematical Monthly.
I want to continue that proofs of wrong statements are just
one kind of useful ways to puzzle children. That they enjoy
it is not astonishible if we remember that children love
performances of magicians, mysteries, fairy tales, `science'
fiction etc. But all this is just a preparation. What should
come next is training in solving problems right - that is so
that to avoid all these `miracles'. That is why it is so
important for children to solve problems where it is clear how
to check the answer and to find out whether it is right or wrong.
Andre Toom
Department of Mathematics toom@universe.iwctx.edu
University of the Incarnate Word Tel. 210-646-0500 (h)
4301 Broadway 210-829-3170 (o)
San Antonio, Texas 78209-6318 Fax 210-829-3153
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