Question

The line joining the mid-points of two chords of a circle passes through its centre. Prove that the chords are parallel.

Answer 1

$E$ is midpoint of $AB$ and $F$ is midpoint of $CD$

Given $EF$ passest hrough $O$ (center)

We know that $OE$ is perpendicular to $AB$ (line drawn from center to midpoint of chord is perpendicular to the chord)

$OE\perp AB$

Similarly $OF\perp CD$

$\angle OEB=\angle OEA={90}^o$ and $\angle OFD=\angle OFC={90}^o$

$EF$ is transversal for $AB$ and $CD$ and $\angle BEF=\angle CFE={90}^o$

interior opposite angles of a tranversal are equal $AB$ is parallel to $CD$

$AB$ and $CD$ are parallel