geometry
A circle is inscribed in trapezoid $ABCD$. Let $K,L,M,N$ be the points of intersection of the circle with diagonals $AC$ and $BD$ respectively. $K$ is between $A$ and $L$ and $M$ is between $B$ and $N$. Given that $AK*LC = 16$ and $BM*ND = \frac{9}{4}$, find the radius of the circle
I was able to deduce a few elementary things
$AB + CD = AD + BC$ and also tried using power of point and use the given products but didnt get anything useful
Let the radius of the circle be $r$; then the height of the isosceles trapezoid is $2r$, and the length of a lateral side would be $4r$.
The four right triangles with $OB$ and $OC$ as hypotenuses are congruent. The four right triangles with $OA$ and $OD$ as hypotenuses are also congruent. Therefore the lengths marked $x$ are all the same, as are those marked $y$.
The area of a trapezoid is $$\frac{1}{2}h(a+b)$$
Therefore $$S = \frac{1}{2}h(a+b)$$
$$S = \frac{1}{2}(2r)(2x + 2y)$$
$$S = 2r(x + y)$$
But we know $x + y = 4r$. Therefore,
$$S = 2r(4r)$$
$$\therefore r = \sqrt{\frac{S}{8}} = \frac{\sqrt{2}}{4}\sqrt{S} \approx 0.3536 \sqrt{S}$$
Note that the triangles $\Delta ABD$ and $\Delta CBD$ are isosceles in $A$ and $C$. This implies that the side bisector of the given diagonal $BD$ is also the median, and angle bisector in the two triangles. It follows that the given quadrilateral has perpendicular diagonals $AC$, $BD$, they intersect in $O$, the mid point of the segment $BD$. Now we consider the triangles $\Delta AOB$ and $\Delta COB$, both with a right angle in $O$.
The first triangle has the known sides $13$, $12$, so $AO=5$.
The second triangle has the known sides $20$, $12$, so $CO=16$.
So the diagonals of the triangle are $BD=24$, and $AC=AO+OC=5+16=21$.
The area of the quadrilateral is half of the area of the rectangle with sides parallel to the diagonals $AC,BD$, where the sides are passing through $A,B,C,D$, so it is $\frac 12 24\cdot 21=12\cdot 21=252$.
Let now $K\in AC$ be the center of the circle which is tangent to all sides.
Let $r$ be the radius of this circle. We consider the four triangles built with $K$ as one vertex, and with one side of the quadrilateral, then compute their areas. We obtain:
$$
\begin{aligned}
252
&=\operatorname{Area}(ABCD)
\\
&=
\operatorname{Area}(\Delta KAB) +
\operatorname{Area}(\Delta KBC) +
\operatorname{Area}(\Delta KCD) +
\operatorname{Area}(\Delta KDA)
\\
&=\frac 12 r(AB+BC+CD+DA)
\\
&=r(AB+BC)
\\
&=33r
\ .
\end{aligned}
$$
This leads to $r=252/33=7.63636363\dots$ .
The closest integer is $8$.
EDIT: Added picture:
Best Answer
When all else fails, get a bigger hammer
...which I did and found that the answer was $R=6$.
I won't explain why $AC$, $BD$ $PR$ and $QS$ concur at $F$. It's just a special case of Brianchon's theorem. So let's start from there. Notice the angles $\angle CAB=\angle ACD=\alpha$, $\angle ABD=\angle BDC=\beta$ and segment $OF=x$. We'll use them all the time. For the sake of simplicity I will also introduce the following symbols: $AK=p_1$, $LC=p_2$, $BM=q_1$, $DN=q_2$. We know that $p_1p_2=16$ and $q_1q_2=9/4$.
First, let's try to find $p_1=AK$. The same approach will be used to find segments $p_2,q_1,q_2$
Take a look at quadrilateral $AQOK$:
$$AQ=FQ\cot\alpha=AK\cos \alpha+OK\sin\varphi$$
$$QO=AK\sin\alpha+OK\cos\varphi$$
This leads to the following equations:
$$p_1\cos\alpha+R\sin\varphi=(R+x)\cot\alpha$$ $$p_1\sin\alpha+R\cos\varphi=R$$
or:
$$R\sin\varphi=(R+x)\cot\alpha-p_1\cos\alpha$$ $$R\cos\varphi=R-p_1\sin\alpha$$
Square these two equations and add them. This will eliminate the angle $\varphi$ that we don't care about. You will end up with a quadratic equation with respect to $p_1$:
$$R^2=((R+x)\cot\alpha-p_1\cos\alpha)^2+(R-p_1\sin\alpha)^2$$
This equation has two solutions. The bigger one is actually equal to $AL$ and the smaller one is $AK=p_1$:
$$p_1=-\sqrt{R^2-\frac{1}{2} x^2 (1+\cos (2 \alpha))}+(R+x)\cos \alpha \cot \alpha +R \sin \alpha\tag{1}$$
There is no need to calculate $p_2$ in a similar way. Everything would look the same, except that we would use $R-x$ instead of $R+x$ along the way. So we can obtain an expression for $p_2$ just by replacing $x$ with $-x$:
$$p_2=-\sqrt{R^2-\frac{1}{2} x^2 (1+\cos (2 \alpha))}+(R-x)\cos \alpha \cot \alpha +R \sin \alpha\tag{2}$$
Now comes the hardest part: you have to multiply (1) and (2). It looks like we are going nowhere... After a lot of simplifications, we get:
$$p_1p_2=-\frac{1}{2} \csc ^2\alpha \left(\left(R^2+x^2\right)\cos (2 \alpha) +2 R \sin \alpha \sqrt{4 R^2-2 x^2 (1+\cos (2 \alpha))}-3 R^2+x^2\right)\tag{3}$$
At least we know it's 16 :)
The good thing is that we don't have to go through the same torture to find $q_1q_2$, the process is exactly the same except that we have $\beta$ instead of $\alpha$:
$$q_1q_2=-\frac{1}{2} \csc ^2\beta \left(\left(R^2+x^2\right)\cos (2 \beta) +2 R \sin \beta \sqrt{4 R^2-2 x^2 (1+\cos (2 \beta))}-3 R^2+x^2\right)\tag{4}$$
Like it's not difficult enough we have to introduce the fact that $x$, $\alpha$ and $\beta$ are not independent! There's a connection between them.
Take a look at triangle $\triangle ABC$:
$$AB=AQ+QB=(R+x)(\cot\alpha+\cot\beta)$$
$$BC=BR+RC=BQ+CS=(R+x)\cot\beta+(R-x)\cot\alpha$$
$$AC={2R \over \sin\alpha}$$
If you replace all that into a well-known equation:
$$BC^2=AB^2+AC^2-2 AB\cdot AC\cos\alpha$$
...you will get a very simple relation (trust me):
$$x^2=R^2(1-\tan\alpha\tan\beta)\tag{5}$$
Now replace (5) into (3) and (4) and we'll get something that we can easily deal with:
$$p_1p_2=R^2\left(2+\frac{\tan\beta}{\tan\alpha}-2\sqrt{1+\frac{\tan\beta}{\tan\alpha}}\right)$$
$$q_1q_2=R^2\left(2+\frac{\tan\alpha}{\tan\beta}-2\sqrt{1+\frac{\tan\alpha}{\tan\beta}}\right)$$
...or, if we introduce $z=\frac{\tan\alpha}{\tan\beta}$:
$$p_1p_2=R^2(2+z-2\sqrt{1+z})=R^2(\sqrt{1+z}-1)^2=16\tag{6}$$
$$q_1q_2=R^2\left(2+\frac1z-2\sqrt{1+\frac1z}\right)=R^2\left(\sqrt{1+\frac1z}-1\right)^2=\frac{9}{4}\tag{7}$$
Now divide (6) by (7) and you get:
$$\left(\frac{\sqrt{1+z}-1}{\sqrt{1+\frac1z}-1}\right)^2=\frac{p_1p_2}{q_1q_2}=\frac{16}{\frac94}=\frac{64}{9}$$
$$\frac{\sqrt{1+z}-1}{\sqrt{1+\frac1z}-1}=\frac{8}{3}$$
$$3(\sqrt{1+z}-1)=8\left(\sqrt{1+\frac1z}-1\right)\tag{8}$$
You should be able to show that equation (8) has only one solution:
$$z=\frac{16}9$$
Replace that into (6) and you get:
$$R^2=\frac{p_1p_2}{(\sqrt{1+z}-1)^2}=36\implies R=6$$