In triangle $ABC$, the perpendicular to the angle bisector from $A$ through the midpoint $A_0$ of side $BC$ intersects the rays $AB$ and $AC$ at points $D_1$ and $D_2$, respectively. The chord $E_1E_2$ of the circle with diameter $D_1D_2$, which is perpendicular to $D_1D_2$ and passes through $A_0$, is to be examined in terms of its size relationship with $BC$!
We can observe that $D_{1} D_{2} \geq E_{1} E_{2}$, since $E_{1} E_{2}$ is a chord in the circle with diameter $D_{1} D_{2}$; if $A B = A C$, then equality holds.

# 1985-03-111-1.eps 

Furthermore, let $A C > A B$. Draw a line parallel to $D_{1} D_{2}$ through point $B$, and let it intersect side $A C$ at $G$, and the angle bisector from $A$ at $F$! Then the quadrilateral $D_{1} D_{2} G B$ is a symmetric trapezoid. Since $F$ bisects $B G$, and $A_{0}$ bisects $B C$, it follows that $A_{0} F \parallel D_{2} G$. Thus, $D_{1} A_{0} F B$ is also a symmetric trapezoid, so its diagonals are equal: $B A_{0} = D_{1} F$. The reflection of the latter segment over the angle bisector from $A$ is the segment $D_{2} F$, so $D_{1} F = D_{2} F$. From the triangle inequality:

$$
D_{1} F + D_{2} F \geq D_{1} D_{2}
$$

Summarizing the above:

$$
\begin{array}{r}
B C = 2 B A_{0} = 2 D_{1} F = D_{1} F + D_{2} F \geq D_{1} D_{2} \geq E_{1} E_{2} \\
B C \geq E_{1} E_{2}
\end{array}
$$

Equality holds precisely when $F$ lies on the line $D_{1} D_{2}$, or equivalently, when $B$ and $C$ do, which means $A B = A C$, and the triangle is isosceles.

Remark. We point out the background to the question. The easily obtained inequality means that if we take the rotational cone whose axis is the angle bisector and one of its generators is the line $A B$, and intersect it with the plane perpendicular to the plane $A B C$ and passing through $B C$, then the segment $B C$ is the major axis of the resulting ellipse, $A_{0}$ is its center, and the segment $E_{1} E_{2}$ gives the length of the minor axis. $D_{1} D_{2}$ is the diameter of the circle that is the intersection through $A_{0}$ and perpendicular to the axis.