document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=524163>Question 524163</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the coordinates of the centroid, orthocenter, and circumcenter of a triangle with vertices A(4, -1), B(2, 6), and C(9, -5).<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #347638 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=347638\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<i><b>Centroid:</b></i>\r<BR>\n" );
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document.write( "Calculate the coordinates of the mid-points of AC and BC. \r<BR>\n" );
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document.write( "Use the mid-point formulas:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}\">\r<BR>\n" );
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document.write( "Write an equation for a line that passes through point A and the mid-point of BC.\r<BR>\n" );
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document.write( "Use the two-point form of an equation of a line:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ \left(\frac{y_1\ -\ y_2}{x_1\ -\ x_2}\right)(x\ -\ x_1) \">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_1,y_1\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_2,y_2\right)\"> are the coordinates of the given points.\r<BR>\n" );
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document.write( "Then write an equation for a line that passes through point B and the mid-point of AC\r<BR>\n" );
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document.write( "Using any convenient method, solve this 2X2 system.  The solution set will be the centroid of the triangle -- the point of intersection of the three triangle medians.\r<BR>\n" );
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document.write( "<i><b>Orthocenter:</b></i>\r<BR>\n" );
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document.write( "Calculate the slope of sides AB and BC of the triangle using the slope formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ m\ =\ \frac{y_1\ -\ y_2}{x_1\ -\ x_2} \">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_1,y_1\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_2,y_2\right)\"> are the coordinates of the given points. \r<BR>\n" );
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document.write( "Then, using the point-slope form of an equation, and the fact that perpendicular lines have slopes that are negative reciprocals, write equations of the two altitudes to sides AB and BC -- lines perpendicular to AC and passing through B and perpendicular to BC and passing through A.\r<BR>\n" );
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document.write( "Solve the 2X2 system.  The intersection of the two altitudes is the orthocenter.\r<BR>\n" );
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document.write( "<i><b>Circumcenter</b></i>\r<BR>\n" );
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document.write( "Using the slopes calculated above for AB and BC and the mid-points calculated for the Centroid solution, write equations of the perpendicular bisectors of AB and BC.  Perpendicular to AB and passing through the mid-point of AB, then perpendicular to BC and passing through the mid-point of BC.\r<BR>\n" );
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document.write( "Solve the 2X2 system.  The intersection of the perpendicular bisectors is the circumcenter (a point equidistant from the three vertices and therefore the center of a circle that passes through the three vertices of the triangle.)\r<BR>\n" );
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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" );
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