Question 1
Let $ABCD$ be a convex quadrilateral. Points $E,F,G,H$ lie on $AB,BC,CD,DA$, respectively, such that lines $AB,FH,CD$ are concurrent, and lines $BC,EG,AD$ are concurrent. Put $O=EG\cap FH$. Consider $AHOE,BEOF,CFOG,DGOH$. Prove that if three of these quadrilaterals are tangential, so is the fourth. A tangential quadrilateral has a circle inside it tangent to all four sides.
