Question 66
Let $ABC$ be an isosceles triangle with $|AB| = |AC|$. The points $B'$ and $C'$ are on the line $BC$ (extended in both directions) such that $B$ is between $B'$ and $C$ and $C$ is between $B$ and $C'$. Moreover, $B'BA$ and $ACC'$ are isosceles triangles. Let $X$ be a point on the line $BC$ such that $B'$ is not between $X$ and $B$.
Prove that $|AX| = |BX|$ if and only if $\frac{1}{|AB|} = \frac{1}{|B'X|} + \frac{1}{|BC'|}$.
Prove that $|AX| = |BX|$ if and only if $\frac{1}{|AB|} = \frac{1}{|B'X|} + \frac{1}{|BC'|}$.