The line segments $AB$ and $CD$ are perpendicular and intersect at point $X$. Additionally, the following equalities hold: $AC=BD$, $AD=BX$, and $DX=1$. Determine the length of segment $CX$.
Source: 2022 Polish Junior Math Olympiad Second Round
Tags: geometr, geometry
The line segments $AB$ and $CD$ are perpendicular and intersect at point $X$. Additionally, the following equalities hold: $AC=BD$, $AD=BX$, and $DX=1$. Determine the length of segment $CX$.