[ $\quad\left[\begin{array}{l}\text { Angle between the tangent and the chord } \\ \text { Auxiliary similar triangles }\end{array}\right]$ [ Trapezoids (other). $\quad]$

In trapezoid $A B C D$ with bases $A D$ and $B C$, diagonals $A C$ and $B D$ intersect at point $E$. A circle is circumscribed around triangle $E C B$, and a tangent to this circle, drawn at point $E$, intersects line $A D$ at point $F$ such that points $A, D$, and $F$ lie sequentially on this line. It is known that $A F=a, A D=$ b. Find $E F$.
Prove the similarity of triangles $D E F$ and $E A F$.

## Solution

The angle vertical to angle $D E F$ is equal to angle $B C E$, so $\angle C A D = \angle B C E = \angle D E F$.

Therefore, triangles $D E F$ and $E A F$ are similar by two angles. Hence, $D F: E F = E F: A F$, from which $E F^{2} = D F \cdot A F = a(a-b)$.

## Answer

$$
\sqrt{a(a-b)}
$$

## [ Angle between a tangent and a chord $\quad]$ Problem 54817 Topics: [ Auxiliary similar triangles ] $[$ Trapezoids (other). $]$

The lateral sides $A B$ and $C D$ of trapezoid $A B C D$ intersect at point $K$. A circle is circumscribed around triangle $B C K$, and a tangent to this circle, drawn at point $K$, intersects line $A D$ at point $L$. It is known that $L K = a, A D = b$. Find $A L$, if $B C < A D$.

## Hint

Prove the similarity of triangles $D K L$ and $K A L$.

## Answer

$\frac{\sqrt{b^{2}+4 a^{2}}+b}{2}$