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### geometry

Posted: **Wed Nov 04, 2015 8:56 pm**

by **Mahfuz Sobhan**

In $$ΔABC$$, $$∠B = 90$$. A circle is drawn taking $$AB$$ as a chord. $$O$$ is the center of the circle. $$O$$ and $$C$$ isn't on the same side of $$AB$$. $$BD$$ is perpendicular to $$AC$$. Prove that, $$BD$$ will be a tangent to the circle if and only if $$∠BAO = ∠BAC$$.

### Re: geometry

Posted: **Sat Nov 21, 2015 8:18 pm**

by **Mallika Prova**

its enough to prove that $\angle BAO=\angle BAC$ when $BD$ is a tangent to the circle...

now,if $BD$ is a tangent $\angle OBD=\angle OBA+\angle ABD=90$.

again,$\angle D=90.\angle BAD+\angle ABD=90$.

then,$\angle OBA=\angle CAB$ and $\angle BAO=\angle BAC$ as,$OB=OA$.

### Re: geometry

Posted: **Sun Nov 22, 2015 1:18 am**

by **sowmitra**

$BD$ will be a tangent $\Leftrightarrow OB\perp BD \Leftrightarrow OB||AC \Leftrightarrow \angle ABO=\angle BAC \Leftrightarrow \angle BAO=\angle BAC$

### Re: geometry

Posted: **Thu Feb 21, 2019 11:47 pm**

by **samiul_samin**

Mahfuz Sobhan wrote: ↑Wed Nov 04, 2015 8:56 pm

In $$ΔABC$$, $$∠B = 90$$. A circle is drawn taking $$AB$$ as a chord. $$O$$ is the center of the circle. $$O$$ and $$C$$ isn't on the same side of $$AB$$. $$BD$$ is perpendicular to $$AC$$. Prove that, $$BD$$ will be a tangent to the circle if and only if $$∠BAO = ∠BAC$$.

This is BdMO National 2014 Secondary P4 & Higher Secondary P3.