[ Isosceles, Inscribed, and Circumscribed Trapezoids Auxiliary Similar Triangles

An isosceles trapezoid $A B C D$ is circumscribed around a circle. The lateral sides $A B$ and $C D$ touch the circle at points $M$ and $N$, and $K$ is the midpoint of $A D$.

In what ratio does the line $B K$ divide the segment $M N$? #
Notice that $M N \| A D$.

## Solution

Let $F$ be the midpoint of $B C$, $Q$ be the intersection of $K F$ and $M N$, and $P$ be the intersection of $M N$ and $B K$. Then $A M=A K$, $B M=B F$, and $Q$ is the midpoint of $M N$.

Since $M N \| A D$, triangles $B M P$ and $B A K$ are similar, as are triangles $K P Q$ and $K B F$. Therefore,

\[
\frac{P M}{A K} = \frac{B F}{A K + B F}, \quad \frac{P Q}{B F} = \frac{A K}{A K + B F}.
\]

Thus, $P M = P Q = \frac{1}{4} M N$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-33.jpg?height=405&width=734&top_left_y=1464&top_left_x=662)

## Answer

$1: 3$.