When is a tangential quadrilateral a kite? (Q2874410)

A convex quadrilateral is called a kite if one of its diagonals divides it into two triangles that are isosceles with the same base. It is called tangential (or circumscriptible) if it admits an incircle (i.e., a circle that touches the sides internally). The paper under review is concerned with characterizing kites among tangential quadrilaterals. Among other things, it proves that if \(ABCD\) is a tangential quadrilateral, and if the incircle touches the sides \(AB\), \(BC\), \(CD\), and \(DA\) at \(X\), \(Y\), \(Z\), and \(W\), respectively, then it is a kite if and only if (i) \(XZ=YW\), (ii) \(AC \perp BD\), (iii) \((AB)(CD) = (AD)(BC)\), (iv) the incenter lies on the longest diagonal, or (v) area (\(ABCD\)) = \((AC)(BD)/2\). It also proves that if \(ABCD\) is a tangential quadrilateral whose diagonals intersect at \(P\), and if \(r_{AB}\) and \(R_{AB}\) are the radii of the incircle and the \(P\)-excircle of triangle \(PAB\) (with similar definitions pertaining to \(PBC\), \(PCD\), \(PDA\)), then \(ABCD\) is a kite if and only if (vi) \(r_{AB}+r_{CD}=r_{AD}+r_{BC}\), (vii) \(r_{AB} r_{CD}= r_{AD} r_{BC}\), (viii) \(R_{AB}+R_{CD}=R_{AD}+R_{BC}\), or (ix) \(R_{AB} R_{CD}=R_{AD} R_{BC}\). The paper gives a new proof of a rather obscure formula that expresses the area of a tangential quadrilateral in terms of the lengths of the sides and the diagonals.NEWLINENEWLINEThe paper is a continuation of a series of papers that the author has written on characterizations of (i) tangential quadrilaterals in [ibid. 11, 65--82 (2011; Zbl 1222.51011); ibid. 12, 63--73 (2012; Zbl 1247.51011)], (ii) orthodiagonal quadrilaterals in [ibid. 12, 13--25 (2012; Zbl 1242.51009)], (iii) rectangles in [ibid. 13, 17--21 (2013; Zbl 1404.51018)], and (iv) trapezoids in [ibid. 13, 23--35 (2013; Zbl 1300.51007)]. More characterizations of tangential quadrilaterals by the same author have recently appeared in [ibid. 14, 1--13 (2014; Zbl 1293.51010)], and several other related papers by the same author, and by others, in the same journal have appeared in the past few years.