3. A circle is inscribed in an isosceles trapezoid $A B C D$ and touches the base $C D$ at point $L$, and the legs $B C$ and $A D$ at points $K$ and $M$, respectively. In what ratio does the line $A L$ divide the segment $M K$?
3. Let the inscribed circle touch the base $AB$ at point $N$. Denote $x=AM=AN=BN=BK$ and $y=CK=CL=DL=DM$. Since $\triangle A X M \sim \triangle A L D$, we have $\frac{M X}{y}=\frac{M X}{D L}=\frac{A M}{A D}=\frac{x}{x+y}$, i.e., $M X=\frac{x y}{x+y}$; since $\triangle L X Q \sim \triangle L A N$, by applying Thales' theorem, we have $\frac{X Q}{x}=\frac{X Q}{A N}=$ $\frac{X L}{A L}=\frac{M D}{A D}=\frac{y}{x+y}$, i.e., $X Q=\frac{x y}{x+y}=M X$. It follows that $M X=\frac{1}{2} M Q=\frac{1}{4} M K$, i.e., $M X: X K=1: 3$.