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On the inscribed quadrilateral again

Summary/Abstract: In the form of problems, various constructions in the plane are considered in the first part of the present paper. The constructions contain quadrilaterals and it is proved that they are inscribed in circles. The proofs are realized by means of the well known criteria for inscribed quadrilateral from the 8th grade curriculum in Geometry. The second part considers application of the inscribed quadrilateral properties to solve geometric problems with different difficulty. One of them treats the well-known property of the symmetric images of the orthocenter of the triangle with respect to the sides and their midpoints. The known Miquel’s theorem for triangles is included in another problem, while one of the next problems is with increased difficulty. It is original. The third part of the paper considers popular properties of the inscribed quadrilateral. Two of them are connected with the angular bisectors of the angles between the opposite sides of the quadrilateral. They are quite interesting. The forth part considers properties of the inscribed quadrilateral under additional assumptions. One of them is a property of the inscribed quadrilateral with perpendicular diagonals. It is known as Brahmagupta’s problem. Another one is a property of the so called harmonic quadrilateral. It is applied to the solution of an interesting problem for a convex hexagon. The fifth part of the paper considers properties of a notable point of the inscribed quadrilateral – its orthocenter. New properties are added to already known. The sixth part considers some known classic theorems, which are connected with the inscribed quadrilateral. Two recently known properties of the gravity center of the inscribed quadrilateral are presented. One of them is classic, while the other one completes the well-known Brocard’s theorem.

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