Problem — Secant to two concentric circles

Text

Two concentric circles have centre $O$. A line meets the larger circle at the points $A$ and $D$ and the smaller circle at the points $B$ and $C$, arranged in the order $A$, $B$, $C$, $D$ along the line. Prove that the segments $AB$ and $CD$, lying between the two circles, are congruent.

Solution

Let $H$ be the foot of the perpendicular drawn from $O$ to the secant line. The perpendicular from the centre to a chord is its axis: hence $H$ is the midpoint of the chord $AD$ of the larger circle and, at the same time, the midpoint of the chord $BC$ of the smaller circle. Therefore $\overline{AH} = \overline{HD}$ and $\overline{BH} = \overline{HC}$. Since $B$ and $C$ lie inside the segment $AD$, $\overline{AB} = \overline{AH} - \overline{BH}, \qquad \overline{CD} = \overline{HD} - \overline{HC} = \overline{AH} - \overline{BH}.$ Therefore $\overline{AB} = \overline{CD}$. $\blacksquare$