Question

Prove that the tangent at the extremities of any chord makes equal angles at the chord.

Answer 1

Let $AB$ be a chord of a circle with centre $O$, and let $AP$ and $BP$ be the tangents at $A$ and $B$ respectively.

Suppose the tangents meet at $P$. Join $OP$. Suppose $OP$ and meets $AB$ at $C$.

We have to prove that $\angle PAC=\angle PBC$.

In triangle $PCA$ and $PCB$, we have

$\Rightarrow$$PA=PB$ [$\because$ tangents from an external point are equal]
$\Rightarrow$$\angle APC=\angle BPC$ [$\because PA$ and $PB$ are equal inclined to $OP$].
$\Rightarrow$$PC=PC$

So, by $SAS------$criterion of congruence, we have

$△PAC\cong △PBC$
$\Rightarrow \angle PAC=\angle PBC$