From: xpolakis@hol.gr (Antreas P. Hatzipolakis)
Subject: The Square Construction Problem
Date: 22 May 1999 11:22:01 GMT
Newsgroups: sci.math
Problem:
Construct a square so that its sides pass through four given
points A,B,C,D.
Analysis:
Let EZHQ be the square in question, whose the sides EQ, EZ, ZH, QH
pass through the given points A,B,C,D (respectively), which no three of
them lie in the same line. We construct AC, and DS _L AC (S lies on EZ).
Also, DN _L EZ, AM _L ZH (N lies on EZ; M on ZH).
The right triangles DNS and AMC are congruent, since:
AM = EZ = ZH = DN, and the acute ang.NDS = ang.MAC (since their sides
are _L ). Therefore DS = AC. Therefore the point S is known, and lies
on EBZ. That is, the direction of EBSZ is known.
Synthesis:
We construct Da _L AC, and take segment DS = AC.
We construct SB, and Db // SB. Then we construct QAE _L SB (Q on Db;
E on SB) and HCZ _L SB (H on Db; Z on SB).
The quadrilateral EZHQ is the square in question.
Apodeixis (well... I mean Proof :-)
EZHQ is rectangular by construction. If we construct AM _L ZH (M on ZH),
DN _L EZ (N on EZ), the right triangles DNS and AMC are congruent, as
having DS = AC (by construction) and angADS = angMAC (their sides are _L ).
Therefore EZ = AM = DN = ZH. Therefore EZHQ is a square.
Diereunesis (in English?):
Let S_1 be on DS so that DS_1 = DS (: S_1 is the symmetrical of S)
We construct S_1B, and Dc // S_1B.
From both A and C we construct _L _L to S_1B and Dc.
The square E_1Z_1H_1Q_1 we get is another solution (2nd).
We construct Dd _L AB, and take on it the segments DS_2 = DS_3 = AB.
We construct CS_2, and De // CS_2. From both A and B we construct
_L _L to CS_2 and De. The square E_2Z_2H_2Q_2 we get is another one
solution (3rd)
We construct CS_3, and Df // CS_3. From both A and B we construct
_L _L to CS_3 and Df. The square E_3Z_3H_3Q_3 we get is another
solution (4th)
Now, we construct Dg _L BC. We take on Dh the segments DS_4 = DS_5 = BC.
We construct AS_4, and Dh // AS_4. From both B and C we construct
_L _L to AS_4 and Dh. The square E_4Z_4H_4Q_4 we get is another one
solution (5th)
We construct AS_5, and Di // AS_5. From both B and C we construct
_L _L to AS_5 and Di. The square E_5Z_5H_5Q_5 we get is another
solution (6th)
Notes:
If happens to be S = B, then DB = _L AC, and there are infinitely many
solutions.
The case: [A,B,C,D are collinear] has an interesting solution I leave
to the reader. There are 12 solutions in general (6 + 6 symmetrical).
This solution was published by Ioannis F. Panakis in a Greek
periodical (1966)
Hope you understand my poor English.
Antreas
==============================================================================
From: kubo@adams.math.brown.edu (Tal Kubo)
Subject: Re: A construction problem
Date: 19 Jan 1999 13:56:46 -0500
Newsgroups: sci.math
Keywords: Citation: given four points construct a square through them
Mike McCarty wrote:
>)>C. A. M. Wildhagen wrote:
>)>)reference for the following problem: [square through 4 points]
It is solved in vol.1 or 2 of Yaglom GEOMETRIC TRANSFORMATIONS, and
also in one of the Kurschak HUNGARIAN PROBLEM BOOKs. Both are
published by the MAA.
>[...] Connect G and H. GH is parallel
>to AB, and is the same length as EF, EG, and FG.
GH doesn't contain A B C or D, so this doesn't answer the question.
The actual problem is nontrivial.