Example 1.1.2 (CMO 1) In $\triangle A B C$, the altitude $A D$ on side $B C$ is $12$, the angle bisector $A E$ of $\angle A$ is $13$, and let the median $A F$ on side $B C$ be $m$. For what range of values of $m$ is $\angle A$ acute, right, or obtuse?

Question 1 In $\triangle A B C$, the altitude $A D$ on side $B C$ is $12$, the angle bisector $A E$ of $\angle A$ is $13$. Let the median $A F$ on side $B C$ be $m$. When $\angle A$ is a right angle, what is the value of $m$?
Solve for $\angle A=90^{\circ}$, $\angle B A D=\angle C=\angle C A F, \angle D A E=\angle E A F, \angle D A F=2 \angle D A E$. At this time,
$$
m=A F=\frac{A D}{\cos \angle D A F}=\frac{12}{\cos \left[2 \arccos \frac{12}{13}\right]}=\frac{12}{2 \cdot\left(\frac{12}{13}\right)^{2}-1}=\frac{2028}{119} .
$$

First, let's explain the positional relationship of points $D$, $E$, and $F$ on line $B C$: If $A B A E$, i.e., $m>13$.

Draw a perpendicular from point $F$ to $B C$, intersecting the arc of the circumcircle of $\triangle A B C$ that does not contain point $A$ at point $G$. Connect $A G$, which is the angle bisector of $\angle B A C$. At this time, consider any segment of the circle, the maximum and minimum distances from the midpoint of the segment's chord to any point on the segment's arc are the height of the segment and half the length of the segment's chord, respectively. This makes it easy to determine:
Acute angle
$$
\begin{array}{l} 
\text { Acute angle } \\
\angle B A C \text { is a right angle } \\
\text { Obtuse angle }
\end{array} \begin{array}{l}
F G A F .
\end{array}
$$

Next, make an intuitive examination and calculation by drawing a diagram: Construct $\triangle D A E$ such that $A D=12, A E=13, \angle A D E=90^{\circ}$. On both sides of $A E$, draw two rays $A B_{K}$ and $A C_{K}$ such that $\angle B_{K} A E=\angle E A C_{K}=\alpha_{K}$. Let
$$
013$, solve for the range of $m$, and we have:
When $\angle A$ is acute, $132028 / 119$.
Explanation: It can be proven that the necessary and sufficient condition for two distinct line segments $m$, $t$, and $h$ to be the median, angle bisector, and altitude of the same side of a triangle is $m>t>h$. In this case, the angle bisector must lie within the angle formed by the median and the altitude.

Given $m$, $t$, and $h$, a triangle can be determined, and thus $m$, $t$, and $h$ can be used to express the length of side $B C$; and the diameter of the circumcircle of the triangle, etc.

It can also be proven that the necessary and sufficient conditions for $\angle \mathrm{A}$ to be acute, right, or obtuse, given by $m$, $t$, and $h$, are: $2 m h^{2}-(m+h) \cdot t^{2}$ is less than, equal to, or greater than 0.