# Thread: Tangent to a circle!

1. ## Tangent to a circle!

From an external point P , two tangents PA & PB are drawn to a circle with center O. Prove that OP is perpendicular bisector of AB.

2. Originally Posted by prantik007
From an external point P , two tangents PA & PB are drawn to a circle with center O. Prove that OP is perpendicular bisector of AB.
Here PA = PB
OA is perpendicular to PA
OB is perpendicular to PB. Hence QAPB is a cyclic quadrilateral.
Angle OAP = ABP = 90 degrees.
If AB cuts OP at D,angle OAD = angle OPB = angle OPA = angle OBD.
Hence you can show that
D is mid point of AB

3. Hello prantik007
Originally Posted by prantik007
From an external point P , two tangents PA & PB are drawn to a circle with center O. Prove that OP is perpendicular bisector of AB.
My thanks to sa-ri-ga-ma for his contribution so far. He is quite correct up to the point where he says:
Hence you can show that
D is mid point of AB
But I'm afraid that this might be a bit of a cop-out.

I think you also need to use the fact that $\angle DBP = 90^o-\angle OBD$. Then use the fact that $\angle BPD = \angle OBD$ to say that:
$\angle DBP = 90^o-\angle BPD$

$\Rightarrow \angle BDP = 90^o$ (angle sum of $\triangle BDP$)
Now we have established that $AB \perp OP$, it's very easy to show that $\triangle$'s $AOD$ and $BOD$ are congruent, and hence that $AD = DB$.

Can you fill in the details now?