If two intersecting chords of a circle make equal angles with diameter passing through their points of intersection, prove that the chords are equal.

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Solution

It is given that $∠ O F B = ∠ O F D$ .

Draw $O P ⊥ A B$ and $O Q ⊥ C D$ .

In $Δ O F P$ and $Δ O F Q$ ,

$F O = F O (c o m m o n)$

$∠ F P O = ∠ F Q O (e a c h 90^{\circ})$

$∠ O F B = ∠ O F D (g i v e n)$

So, by AAS criteria, $Δ O F P ≅ Δ O F Q$ .

By CPCT, $O P = O Q$

Since, the chords equidistant from the centre are equal in length, then,

$A B = C D$

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