document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=806276>Question 806276</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am looking for a formula that will show if coordinates are within or outside a set parallelogram.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #485794 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=485794\">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( "Assume the coordinates of the four vertices are given and that they are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V_1(x_1,y_1)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V_2(x_2,y_2)\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V_3(x_3,y_3)\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V_4(x_4,y_4)\">.  Since coordinate rotation and the labeling of the vertices is arbitrary, we can assume without loss of generality that the following inequalities hold:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_1\ <\ x_2\ <\ x_4\ <\ x_3\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_1\ <\ y_4\ <\ y_2\ <\ y_3\">\r<BR>\n" );
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document.write( "Now write the following four inequalities using your given coordinates:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ <\ \left(\frac{y_2\ -\ y_1}{x_2\ -\ x_1}\right)\left(x\ -\ x_1\right)\ +\ y_1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ <\ \left(\frac{y_3\ -\ y_2}{x_3\ -\ x_2}\right)\left(x\ -\ x_2\right)\ +\ y_2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ >\ \left(\frac{y_3\ -\ y_4}{x_3\ -\ x_4}\right)\left(x\ -\ x_4\right)\ +\ y_4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ >\ \left(\frac{y_4\ -\ y_1}{x_4\ -\ x_1}\right)\left(x\ -\ x_1\right)\ +\ y_1\">\r<BR>\n" );
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document.write( "If a point on a side of the parallelogram is considered \"in\" the parallelogram, then change the inequality operators from strictly less than/greater than, to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \leq\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \geq\">\r<BR>\n" );
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document.write( "Then if the coordinates of a point satisfy ALL FOUR inequalities, the point is in the parabola, otherwise it is not.\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( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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