In triangle $ABC$, $AB=AC$, the midpoint of base $BC$ is $D$, and the vertex opposite $D$ of the square constructed over segment $BD$ is $E$. The circle centered at $D$ with radius $DE$ intersects the line $BC$ at points $F$ and $G$. We draw a line parallel to $BC$ through $A$, and measure a segment equal to $AB$ from $A$ on this line, with endpoint $H$.

Express the difference $FH-GH$ in terms of the sides of triangle $ABC$.
Let $AB = AC = b$ and $BD = a$, so $DF = DG = DE = a\sqrt{2}$. Let $H$ be on the side of line $AD$ containing $G$, and let the projection of $H$ onto line $BC$ be $H'$.

![](https://cdn.mathpix.com/cropped/2024_05_02_8e2e448e7a04b56ab479g-1.jpg?height=230&width=577&top_left_y=270&top_left_x=752)

Thus, $FH > GH$, and $FH' = DF + DH' = DF + AH = b + a\sqrt{2}$ and $GH' = |DH - DG| = |b - a\sqrt{2}|$, so

\[
\begin{gathered}
FH = \sqrt{FH'^2 + H'H^2} = \sqrt{(b + a\sqrt{2})^2 + DA^2} = \\
= \sqrt{\left(b^2 + 2\sqrt{2}ab + 2a^2\right) + \left(b^2 - a^2\right)} = \sqrt{2b^2 + 2\sqrt{2}ab + a^2} = b\sqrt{2} + a
\end{gathered}
\]

since the length of a segment is positive. Similarly,

\[
GH = \sqrt{GH'^2 + H'H^2} = \sqrt{(b - a\sqrt{2})^2 + (b^2 - a^2)} = b\sqrt{2} - a
\]

since $b\sqrt{2} > b > a$.

Therefore, $FH - GH = 2a = BC$.

Remark. According to the above result, the difference $FH - GH$ depends only on the base of the triangle: if we change the position of point $A$ (while $B$ and $C$ are fixed), the difference $FH - GH$ for the constructed point $H$ in any position will be the same. Thus, point $H$ moves along a hyperbola, since $F$ and $G$ are fixed.

Knowing the equation of a hyperbola in a coordinate system, we could have obtained this result (and thus the statement of the problem) more briefly: choose the lines $BC$ and $DA$ as the axes of a coordinate system, then the coordinates $x$ and $y$ of point $H$ (where $H'D = x$ and $H'H = y$) satisfy

\[
x^2 = DH'^2 = AB^2, \quad y^2 = DA^2
\]

so the relation

\[
x^2 - y^2 = BD^2
\]

holds. This is the equation of a hyperbola, whose foci are indeed the points $F$ and $G$ as specified in the problem.