Problems

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Found: 133

a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?

b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?

Prove that the bisectors of a triangle intersect at one point.

Prove that a convex quadrilateral \(ABCD\) can be inscribed in a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).

Prove that a convex quadrilateral \(ICEF\) can have a circle inscribed into it if and only if \(IC+EH = CE+IF\).

The triangle \(ABC\) is given. Find the locus of the point \(X\) satisfying the inequalities \(AX \leq CX \leq BX\).

Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Using only straightedge and compass construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).