Construct quadrilateral $ABCD$: $AB=AD=4$, $CB=CD=5$ and $BD=6$. Construct incircle of quadrilateral $ABCD$.

Question

Its easy to construct $ABCD$ but how can we construct incircle? As $ABCD$ is a kite is there any method? For my efforts I tried incircle as in triangle will it satisfy the question?

Answer 1

Note that the centre of the incircle lies on the angle bisector of angle $ABC$, as well as the angle bisector of $ADC$ and of $BAD$. Since this is a kite, these three lines are concurrent, and furthermore the angle bisector of $BCD$ is the same as that of $BAD$, so you only need construct the bisector of $ABC$ and intersect it with line $AC$. From here, simply drop a perpendicular to $BC$ to give a radius and centre of the desired circle.

Answer 2

For the incircle, the center is at the same distance from $AD$ and $AB$. Therefore it is on the bisector of $\angle DAB$. At the same time, due to symmetry, it is on the $BD$ line (which also happens to be the bisector of $\angle ABC$). Find the intersection. Draw the perpendicular from the center to any of the sides