document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1043916>Question 1043916</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A circle is inscribed in an equilateral triangle. If the circumference of it is 1cm find the exact perimeter of this triangle. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #659124 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=659124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The center of a circle inscribed in a triangle is called the incenter of the triangle.  The incenter is located at the intersection of the bisectors of the three angles of the triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Refer to the diagram\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=CircleInscribedInEquilateralTri_(2).jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "BD bisects angle ABC, AE bisects angle BAC.  Since ABC is equilateral, BD is the perpendicular bisector of AC, and AE is the perpendicular bisector of BC.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "All angles of an equilateral triangle measure 60 degrees, so angle OAD must measure 30 degrees.  Since BD perpendicular to AC, angle ODA measures 90 degrees, then angle AOD measures 60 degrees perforce.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the inscribed circle is tangent at D, D is on the circle, hence OD is a radius of the circle.  Given the circumference is 1 cm, the radius of the circle must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2\pi}\"> cm.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the sides of a 30:60:90 right triangle are in proportion <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{2}:\frac{\sqrt{3}}{2}:1\">,  segment AD must measure <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\sqrt{3}}{2\pi}\"> cm.  Since segment AD is 1/6 of the perimeter of the equilateral triangle, the entire perimeter must be:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 6\,\cdot\,\frac{\sqrt{3}}{2\pi}\ =\ \frac{3\sqrt{3}}{\pi}\"> cm.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );