Problem

Source: Romanian JBTST II 2007, problem 2

Tags: geometry, rectangle, geometry proposed



Consider a convex quadrilateral $ABCD$. Denote $M, \ N$ the points of tangency of the circle inscribed in $\triangle ABD$ with $AB, \ AD$, respectively and $P, \ Q$ the points of tangency of the circle inscribed in $\triangle CBD$ with the sides $CD, \ CB$, respectively. Assume that the circles inscribed in $\triangle ABD, \ \triangle CBD$ are tangent. Prove that: a) $ABCD$ is circumscriptible. b) $MNPQ$ is cyclic. c) The circles inscribed in $\triangle ABC, \ \triangle ADC$ are tangent.