INTERNATIONAL JOURNAL OF GEOMETRY
Vol. 1 (2012), No. 2, 61 - 64
A NEW PROPERTY OF
CIRCUMSCRIBED QUADRILATERAL
MIHAI MICULIŢA
Abstract. In this note we will give a new property of circumscribed
quadrilateral.
1. Introduction
A circumscribed quadrilateral is a convex quadrilateral with an incircle,
that is a circle tangent to all four sides. Figure 1 shows a circumscribed
quadrilateral ABCD; where his incircle touch its sides AB; BC; CD; DA at
the points X; Y; Z; T; respectively.
Other names for these quadrilaterals are tangent quadrilateral, inscriptable quadrilateral and circumscriptible quadrilateral. For more details we
refer to the monograph of D. Grinberg [3] or D. Mihalca, I. Chiţescu and M.
Chiriţ¼
a [9] and to the papers of T. Andreescu and B. Enescu [1], W. Chao
and P. Simeonov [2], M. Josefsson [4], [5], [6], M. Hajja [7], L. Hoehn [8], N.
Minculete [10] and M. De Villiers [11].
————————————–
Keywords and phrases: circumscribed quadrilateral, inversion
(2010)Mathematics Subject Classi…cation: 51M04, 51M25
Received: 21.05.2012. In revised form: 4.06.2012. Accepted: 28.06.2012.
62
Mihai Miculiţa
A convex quadrilateral with the sides a; b; c; d is tangential if and only if
(1)
a+c=b+d
according to the Pitot theorem [1, pp. 65-67].
The following result was obtained by A. Zaslavsky in [3].
In a circumscribed quadrilateral ABCD we note with K; L; M and N
the projections of the intersection point of the diagonals of ABCD on the
[AB] ; [BC] ; [CD] and [AD] sides. The following relation holds:
(2)
1
1
1
1
+
=
+
:
jOKj jOM j
jOLj jON j
2. Main result
Theorem 2.1. Let ABCD be a circumscribed quadrilateral and O is the
point of intersection of its diagonals. Let A1 B1 C1 D1 be a quadrilateral obtained by inversion of pole O of quadrilateral ABCD. Then A1 B1 C1 D1 is
circumscribed quadrilateral.
Proof. Let K; L; M and N be the projections of the point O on the
[AB] ; [BC] ; [CD] and [AD] ; respectively (see Figure 2).
Denote by SOAB the area of the triangle OAB and by k the ratio of the
inversion of pole O: From the equalities
(3)
\ = jABj jOKj ;
2SOAB = jOAj jOBj sin AOB
A new property of circumscribed quadrilateral
63
we obtain
(4)
Similarly, we have
\
jABj
sin AOB
=
jOAj jOBj
jOKj
\
jCDj
sin COD
=
jOCj jODj
jOM j
(5)
\ = sin COD,
\ by (5) and (6) results
Because sin AOB
jA1 B1 j + jC1 D1 j = k
\
= k sin AOB
(6)
Similarly, we have
(7)
jCDj
jABj
+
jOAj jOBj jOCj jODj
1
1
+
jOKj jOM j
\
jA1 D1 j + jB1 C1 j = k sin DOA
1
1
+
jOLj jON j
:
Using the relation (2), we have
1
1
1
1
+
=
+
:
(8)
jOKj jOM j
jOLj jON j
\ = sin DOA,
\ by (6), (7) and (8), we obtain that:
Because sin AOB
(9)
jA1 B1 j + jC1 D1 j = jA1 D1 j + jB1 C1 j :
Now, by (1) and (9) result the conclusion.
References
[1] Andreescu, T. and Enescu, B., Mathematical Olympiad Treasures, Birkhäuser,
Boston, 2004.
[2] Chao, W. and Simeonov, P., When quadrilaterals have inscribed circles (solution to
problem 10698), American Mathematical Monthly, 107(7)(2000), 657–658.
[3] Grinberg, D., Circumscribed quadrilaterals revisited, 2008, pdf.
[4] Josefsson, M., More Characterizations of Tangential Quadrilaterals, Forum Geometricorum 11(2011), 65–82.
[5] Josefsson, M., Calculations concerning the tangent lengths and tangency chords of a
tangential quadrilateral, Forum Geometricorum, 10(2010), 119–130.
[6] Josefsson, M., Similar Metric Characterizations of Tangential and Extangential
Quadrilaterals, Forum Geometricorum, 12(2012), 63-77.
[7] Hajja, M., A condition for a circumscriptible quadrilateral to be cyclic, Forum Geometricorum, 8(2008), 103–106.
[8] Hoehn, L., A new formula concerning the diagonals and sides of a quadrilateral,
Forum Geometricorum, 11(2011), 211–212.
[9] Mihalca, D., Chiţescu, I. and Chiriţ¼
a, M., The quadrilateral’s geometry , Teora,
Bucharest, 1998.
[10] Minculete, N., Characterizations of a Tangential Quadrilateral, Forum Geometricorum, 9(2009), 113–118.
[11] Shkljarskij, D., Chenzov, N. and Jaglom, I., Izbrannye zadachi i teoremy elementarnoj
matematiki: Chastj 2 (Planimetrija), Moscow 1952.
64
Mihai Miculiţa
[12] De Villiers, M., Equiangular cyclic and equilateral circumscribed polygons, Mathematical Gazette, 95(2011), 102–107.
[13] Zaslavsky, A., Problem M.1887, Kvant, 6(2003), Nauka Publisher House, Russia.
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