3. In a convex quadrilateral $A B C D$, the lengths of sides $A B$ and $B C$ are equal, $D B$ is the bisector of angle $A D C$, and $A D: D C=4: 3$. Find the cosine of angle $A K B$, where $K$ is the intersection point of diagonals $A C$ and $B D$, and $B K: K D=1: 3$. (16 points)
# Solution.

1) $A D: D C=4: 3$, let $A D=4 x, D C=3 x$, $B K: K D=1: 3$, let $B K=y, K D=3 y$.
2) $D K$ is the bisector of triangle $A D C$, $A K: K C=A D: D C=4: 3, A K=4 z, K C=3 z$.
3) Point $B$ is the intersection of the perpendicular bisector of diagonal $A C$ and the bisector of angle $D$ in the convex quadrilateral $A B C D$. Therefore, a circle can be circumscribed around this quadrilateral. Indeed, let's circumscribe a circle around triangle $A C D$, and denote the intersection of the bisector of angle $D$ with the circle as $B_{1}$. Then, by the property of inscribed angles, the arcs $A B_{1}$ and $B_{1} C$ will be equal, the chords $A B_{1}$ and $B_{1} C$ will also be equal, triangle $A B_{1} C$ will be isosceles, and the perpendicular bisector of diagonal $A C$ and the bisector of angle $D$ will intersect at point $B_{1}$. Therefore, $B_{1}=B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b3b045c4417b92ef0023g-03.jpg?height=639&width=626&top_left_y=997&top_left_x=1132)
4) Since a circle can be circumscribed around quadrilateral $A B C D$, the following equality holds for its diagonals: $A K \cdot K C=B K \cdot K D, 4 z^{2}=y^{2}, y=2 z$.
5) Triangle $A B K$ is similar to $D C K$, and $\frac{A B}{C D}=\frac{B K}{K C}$, let $A B=p, \frac{p}{3 x}=\frac{y}{3 z}=\frac{2}{3}, p=2 x, x=\frac{p}{2}$.
6) $A D=2 p, D C=\frac{3 p}{2}$.
7) By the cosine theorem for triangles $A B C$ and $A D C$, taking into account $\angle B+\angle D=180^{\circ}$, we have $49 z^{2}=2 p^{2}-2 p^{2} \cos \angle B, 49 z^{2}=4 p^{2}+\frac{9 p^{2}}{4}+6 p^{2} \cos \angle B$. From this, $z=\frac{p}{4}, y=\frac{p}{2}, A K=p, B K=\frac{p}{2}$. 8) For the isosceles triangle $A B K$, we have $\cos \angle A K B=\frac{B K}{2 A K}=\frac{1}{4}$.

Answer: $\mathbf{1 / 4}$.