Selected Problems from Open Contests

a) As the incenter of the triangle $ABC$ is located on diagonal $BD$ (Fig. 1), we can conclude that $BD$ is the bisector of $\angle ABC$. Therefore $\angle ABD=\angle CBD$. However, since $ABCD$ is a parallelogram, $\angle ABD=\angle CDB$. Hence the triangle $BCD$ is isosceles, i.e. $|BC|=|CD|$. Thus, $ABCD$ is a rhombus.

Fig. 1

b) Let $ABCD$ be a rectangle with different side lengths. The circumcenter of triangle $ABC$ is located on the intersection of the diagonals of the rectangle. We see that all the required conditions are satisfied, however $ABCD$ is not a rhombus.