A New Property of Circumscribed Quadrilateral

INTERNATIONAL JOURNAL OF GEOMETRY Vol. 1 (2012), No. 2, 61 - 64

A NEW PROPERTY OF CIRCUMSCRIBED QUADRILATERAL

MIHAI MICULI¸TA

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, inscript- able quadrilateral and circumscriptible quadrilateral. For more details we refer to the monograph of D. Grinberg [3] or D. Mihalca, I. Chi¸tescu and M. Chiri¸ta¼ [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 MihaiMiculi¸ta

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: 1 1 1 1 (2) + = + : OK OM OL ON j j j j j j j j

2. Main result

Theorem 2.1. Let ABCD be a circumscribed quadrilateral and O is the point of intersection of its diagonals. Let A1B1C1D1 be a quadrilateral obtained by inversion of pole O of quadrilateral ABCD. Then A1B1C1D1 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) 2SOAB = OA OB sin AOB\ = AB OK ; j j j j j j j j

Anewpropertyofcircumscribedquadrilateral 63 we obtain AB sin AOB\ (4) j j = OA OB OK j j j j j j Similarly, we have CD sin COD\ (5) j j = OC OD OM j j j j j j Because sin AOB\ = sin COD\, by (5) and (6) results AB CD A1B1 + C1D1 = k j j + j j j j j j OA OB OC OD j j j j j j j j 1 1 (6) = k sin AOB\ + OK OM j j j j Similarly, we have 1 1 (7) A1D1 + B1C1 = k sin DOA\ + : j j j j OL ON j j j j Using the relation (2), we have 1 1 1 1 (8) + = + : OK OM OL ON j j j j j j j j Because sin AOB\ = sin DOA\, by (6), (7) and (8), we obtain that:

(9) A1B1 + C1D1 = A1D1 + B1C1 : j j j j j j j 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 Geomet- ricorum 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 Geo- metricorum, 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., Chi¸tescu, I. and Chiri¸ta,¼ M., The quadrilateral’s geometry, Teora, Bucharest, 1998. [10] Minculete, N., Characterizations of a Tangential Quadrilateral, Forum Geometrico- rum, 9(2009), 113–118. [11] Shkljarskij, D., Chenzov, N. and Jaglom, I., Izbrannye zadachi i teoremy elementarnoj matematiki: Chastj 2 (Planimetrija), Moscow 1952. 64 MihaiMiculi¸ta

[12] De Villiers, M., Equiangular cyclic and equilateral circumscribed polygons, Mathemat- ical Gazette, 95(2011), 102–107. [13] Zaslavsky, A., Problem M.1887, Kvant, 6(2003), Nauka Publisher House, Russia.

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