3. In a right-angled triangle $ABC$, with the right angle at vertex $C$, the ratio of the lengths of the altitude and the median drawn from the vertex of the right angle is equal to 12 : 13. Determine the ratio of the lengths of the legs of the triangle $a: b$ if $a > b$.
First method:

With the notations $|\angle C A B|=\alpha$ and $|\angle A B C|=\beta$, we have $\alpha+\beta=90^{\circ}$. Since triangles ANC and $C N B$ are right triangles, it follows that $|\angle N C A|=\beta$ and $|\angle B C N|=\alpha$.

Therefore, triangles ANC and CNB are similar by the AA similarity criterion, so we have

$\frac{a}{b}=\frac{v_{c}}{\frac{c}{2}-x}=\frac{12 k}{8 k}=\frac{3}{2}=3: 2$. $\qquad$
TOTAL 10 POINTS