[  Theorem of the sum of squares of diagonals  Law of Cosines  [ Area of a triangle (through semiperimeter and radius of inscribed or exscribed circle

In parallelogram $A B C D$, lines $l 1$ and $l 2$ are the angle bisectors of angles $A$ and $C$ respectively, and lines $m 1$ and $m 2$ are the angle bisectors of angles $B$ and $D$ respectively. The distance between $l 1$ and $l 2$ is $\sqrt{3}$ times the distance between $m 1$ and $m 2$. Find the angle $B A D$ and the radius of the circle inscribed in triangle $A B D$, if $A C=3, B D=\sqrt{\frac{5 g}{3}}$.
![](https://cdn.mathpix.com/cropped/2024_05_06_4bea0374a35e8bb7d318g-05.jpg?height=1035&width=1838&top_left_y=0&top_left_x=144)

## Answer

$\frac{2 \pi}{3}, \frac{1}{\sqrt{3}}$.

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