Perpendicular Bisectors in an Inscriptible Quadrilateral: What is this about?
A Mathematical Droodle
What if applet does not run? 
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Copyright © 19962018 Alexander BogomolnyThe applet may suggest the following statement [de Villiers]:
Given a qudrilateral ABCD, the perpendicular bisectors of its sidelines form a quadrilateral A_{1}B_{1}C_{1}D_{1}. If ABCD is inscriptible, then so is A_{1}B_{1}C_{1}D_{1}. 
What if applet does not run? 
The applet is suggestive of a statement but does not seem to help with finding a proof. The theorem could be derived from a similar theorem concerned with angles bisectors. A presentation of the proof with a different illustration appears elsewhere. Below, I give a proof kindly supplied to me by M. de Villiers. The proof is included in his book Some Adventures in Euclidean Geometry and was suggested by Jordan Tabov.
I apologize for a change of notations.
As seen in the diagram, the given quadrilateral ABCD is inscriptible. X, Y, Z, W are the midpoints of the sides AD, AB, BC, and CD, respectively.
We want to show that the quadrilateral EFGH is inscriptible. Let
We now have:
(1) 
2·MD = d/cosD, 2·MW = c  d/cosD, 2·FW = 2·MW·cot(180°  D) = d/sinD  c·cotD. 
Similarly
(2)  2·FX = c/sinD  d·cotD. 
Hence
(3) 

Simialrly for 2(HZ  HY), 2(EY  EX) and
EF  EH + HG  GF = 0, 
which is the same as
(FXEX)  (HYEY) + (HZGZ)  (FWGW) = 0. 
Rearranging the terms and multiplying by 2 gives
(4)  2(FX  FW) + 2(HZ  HY) + 2(EY  EX) + 2(GW  GZ) = 0. 
In view of (3), (4) is equivalent to
(5)  (c  d)t_{D} + (d  a)t_{A} + (a  b)t_{B} + (b  c)t_{C} = 0. 
Since ABCD is inscriptible,
(6)  (c  d)(t_{D}  t_{B}) + (d  a)(t_{A}  t_{C}) = 0. 
But
c  d = (t_{C} + t_{D})  (t_{D}) + t_{A}) = t_{C}  t_{A}. 
and similarly
d  a = t_{D}  t_{B}. 
Therefore (6) is equivalent to
(t_{C}  t_{A})(t_{D}  t_{B}) + (t_{D}  t_{B})(t_{A}  t_{C}) = 0. 
which is an identity and completes the proof.
(A zipped Sketchpad sketch is available for download.)
References
 M. de Villiers, Private communication, Dec. 2004
 M. de Villiers, Some Adventures in Euclidean Geometry, Univ. of DurbanWestville, 1994 (revised 1996), pp. 192193
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny66156731