Triangle ABC has altitudes <i>h</i><sub>1</sub>, <i>h</i><sub>2</sub> and <i>h</i><sub>3</sub>.
<p/>
The radius of the inscribed circle is <i>r</i>, while the radii of the escribed circles 
are <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub> and <i>r</i><sub>3</sub> respectively.
" 





<p>Prove:</p>

<p>



 
  


<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center"><tr><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>r</i><br /></td><td nowrap="nowrap" align="center">
 = </td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>h</i><sub>1</sub><br /></td><td nowrap="nowrap" align="center">
+ </td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>h</i><sub>2</sub><br /></td><td nowrap="nowrap" align="center">
+ </td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>h</i><sub>3</sub><br /></td><td nowrap="nowrap" align="center">
 = </td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>r</i><sub>1</sub><br /></td><td nowrap="nowrap" align="center">
+ </td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>r</i><sub>2</sub><br /></td><td nowrap="nowrap" align="center">
+ </td><td nowrap="nowrap" align="center">
&#x00A0;1
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>r</i><sub>3</sub><br /></td><td nowrap="nowrap" align="center">
.</td></tr></table>
</td><td width="1%">(1)</td></tr></table>


<p/>


</p>

<img width="376" height="269" src="/content/00/05/15plus2/fig1.gif"
alt="Area" /><br />