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Оn the circumscribed quadrilateral again

Summary/Abstract: Various properties of a circumscribed quadrilateral are considered in the first part of the paper. Two of them concern the lines defined by the tangent points of the in-circle with the opposite and respectively with the adjacent pairs of the quadrilateral sides. One of the considered properties of the circumscribed quadrilateral is a corollary of the well-known Brocard theorem for inscribed quadrilaterals. Another one is related to the circles that are inscribed in the two triangles defined by a diagonal of a circumscribed quadrilateral and its non-adjacent vertices. In addition a new discovered property of the circumscribed quadrilateral is considered representing a relation with the participation of the distances between its vertices and the in-center. By means of that relation another interesting one between the distances is derived in the case when the quadrilateral is inscribed in a circle. Also, of interest is one more property of the circumscribed quadrilateral connected with the orthocentres of the triangles defined by its vertices and the in-centre. The second part of the paper is dedicated to some properties of an in-circum scribed quadrilateral. The third one contains problems for circumscribed quadrilaterals from various mathematical contests and exams, while the forth one considers known classic theorems connected with the circumscribed quadrilateral. Newton’s theorem for the circumscribed quadrilateral and an interesting addition to it are presented here.

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