Problem:
In an acute angled triangle $ABC$, $\angle A= 60^0$. Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the circumcircle of the triangle $ABC$.
BdMO National 2012: Higher Secondary, Secondary 07
BdMO National 2012: Higher Secondary, Secondary 07
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Re: BdMO National 2012: Higher Secondary, Secondary 07
what is that "angles formed by the altitudes drawn from $B$ and $C$" ?Moon wrote:Problem:
In an acute angled triangle $ABC$, $\angle A= 60^0$. Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the circumcircle of the triangle $ABC$.
Try not to become a man of success but rather to become a man of value.Albert Einstein
Re: BdMO National 2012: Higher Secondary, Secondary 07
Let the feet of altitudes from $B,C$ be $E,F$. If $BE$ and $CF$ intersects at $H$ then the angles are $\angle BHC,\angle CHE,\angle EHF,\angle BHF$.
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Re: BdMO National 2012: Higher Secondary, Secondary 07
O is center.radius R
\[BC=\frac{2R}{sinA}=\sqrt{3}R\]
$\angle BOC=2x60^O=120^O$ AND $\angle BHC=120^O$. so, $B,H,O,C$ are cyclic.
in tri $BOC$, \[sin \angle OBC=\frac{sinA.R}{BC}=\frac{sin60^O.R}{\sqrt{3}R}=\frac{1}{2}\]
so, $\angle OBC=30^O$, $\angle OHC=30^O$[$B,H,O,C$ are cyclic.]
so, $HO$ bisects $\angle CHE=60^O$.Done.
\[BC=\frac{2R}{sinA}=\sqrt{3}R\]
$\angle BOC=2x60^O=120^O$ AND $\angle BHC=120^O$. so, $B,H,O,C$ are cyclic.
in tri $BOC$, \[sin \angle OBC=\frac{sinA.R}{BC}=\frac{sin60^O.R}{\sqrt{3}R}=\frac{1}{2}\]
so, $\angle OBC=30^O$, $\angle OHC=30^O$[$B,H,O,C$ are cyclic.]
so, $HO$ bisects $\angle CHE=60^O$.Done.
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Try not to become a man of success but rather to become a man of value.Albert Einstein
Re: BdMO National 2012: Higher Secondary, Secondary 07
When the altitudes intersect, 2 acute angles and 2 obtuse angles are formed. If my check is right, then $OH$ always divides the acute ones.
Do we need to prove this?
And also, there is an easy euclidean proof taking this fact into account.
Do we need to prove this?
And also, there is an easy euclidean proof taking this fact into account.
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