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Easy Construction of Bicentric Quadrilateral: What Is This About?
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Copyright © 1996-2010 Alexander Bogomolny
Easy Construction of Bicentric Quadrilateral
A quadrilateral is bicentric if it's both inscriptable and circumscriptable. (Inscriptable means admitting an incircle. Circumscriptable means cyclic, i.e., admitting a circumcircle.) Bicentric quadrilaterals might seem exotic, but the applet shows how such quadrilaterals can be constructed easily. (Another construction appears elsewhere.)
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Let ABCD be a cyclic quadrilateral with vertices on a given circle w. Assume ABCD is also orthodiagonal, i.e., 
BD.
The latter assertion can be rephrased. We know that, the diagonals of a cyclic quadrilateral (ABCD in this case) and those of the quadrilateral formed by the points of tangency of ABCD and its incircle, are concurrent. Therefore we can say that, for a bicentric quadrilateral, the intersecition of the diagonals M, the incenter I and the circumcenter O are collinear.
Proof
First of all note the following angle identities
| (1) |  QCB = BCQ = BAC = BDC = a. | (2) | ADS = DAS = ABD = ACD = b. |
(3) | AEB = CED = g. |
We have to show that the diagonals of ABCD are orthogonal, i.e.,
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PQR + PSR = 180o,
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or, which is the same, iff
| (4) |
BQC + ASD = 180o
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Now, in 
BCQ,
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BQC + 2a = 180o,
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so that from (1)
| (*1) |
BQC + 2BAC = 180o,
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In ADS,
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ASD + 2b = 180o,
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so that from (2)
| (*2) |
ASD + 2ABD = 180o,
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Thus, in ABE,
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which proves the first part of the assertion. The fact that I, O, and E are collinear has been proven in [Dubrovsky] and [Honsberger]. An absolutely delicious proof appears as a consequence of another construction of bicentric quadrilaterals.
References
- G. Bennett, Bi-centric Quadrilaterals and the Pedal n-gon, in The Lighter Side of Mathematics, R.K.Guy and R.E.Woodrow, eds, MAA, 1994, p. 97
- J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971, p. 45
- H. Dorrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965, pp. 188-193
- V. N. Dubrovsky, Solution to problem M1154, Kvant, n 8, 1989, pp 34-35 (in Russian), pdf is available at http://kvant.mccme.ru/1989/08/p34.htm.
- R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960, p. 95
- R. Honsberger, In Pólya's Footsteps, MAA, 1999, pp. 100-101
Copyright © 1996-2010 Alexander Bogomolny
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