Circle Tangents

The Problems :

 A                                       

Let ABCD be a convex quadrilateral (no reflex angles), and suppose the quadrilateral has an inscribed circle, that is, a circle inside the shape that touches all four of the line-segments AB, BC, CD, DA .

 

 

Prove that |AB| + |CD| = |BC| + |DA|.

 Deduce that if a parallelogram has an inscribed circle, then it is a rhombus.  

B  (Harder)  

Let ABCD be a convex quadrilateral such that |AB| + |CD| = |BC| + |DA|. Does it follow that ABCD necessarily has an inscribed circle? Can you find a 'counter-example' ?

 

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Last modified: June 18, 2007