Areal coordinates of the orthocenter

Laplace · New Member Offline Joined: 10 Apr 2009 Posts: 1

I need to show that areal coordinates of H are
H = (cotB cotC, cotA cotC, cotA cotB), where H is the orthocenter of a triangle ABC.

I know that the areal coordinates of point H are:

However I'm not sure how to show that the two are equal.

**Posted:** Fri Apr 10, 2009 10:39 am

luisgeometria

The areal coordinates of a point are the normalized barycentric coordinates:
The barycentric coordinates of $H$ are $(tanA: tanB: tanC)$ ...Normalized $\Longrightarrow$
$\left (\frac {tanA}{tanA + tanB + tanC}: \frac {tanB}{tanA + tanB + tanC}: \frac {tanC}{tanA + tanB + tanC} \right)$

Now use the identity $tanA.tanB.tanC = tanA + tanB + tanC$ and the result follows:

$H: (cotBcotC: cotAcotC;cotBcotC)$

**Posted:** Sat Apr 11, 2009 10:09 am

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