document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200535>Question 1200535</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The line x - 2y + 4 = 0 is tangent to a circle at (0,2). The line y = 2x - 7 is tangent to the same circle at (3, -1). Find the center of the circle.\r<BR>\n" );
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document.write( "NOTE: I WORKED THIS OUT WRONGLY ON PAPER. I DON'T KNOW HOW TO UPLOAD PHOTOS ON THIS MATH SITE WHICH IS VERY LIMITED.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834754 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52790)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=834754\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "This problem has an underwater stone, &nbsp;which is usually unseen to many people.\r<BR>\n" );
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document.write( "This underwater stone is that the problem is OVER-defined.\r<BR>\n" );
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document.write( "Indeed, &nbsp;the condition giving coordinates of &nbsp;TWO &nbsp;tangent points is &nbsp;EXCESSIVE: <BR>\n" );
document.write( "one tangent point is just enough and it defines the second tangent point<BR>\n" );
document.write( "by an &nbsp;UNIQUE &nbsp;way.\r<BR>\n" );
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document.write( "In couple of words, &nbsp;I will explain &nbsp;WHY &nbsp;the problem is over-defined.\r<BR>\n" );
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document.write( "    Indeed, we know that the center must lie on the bisector of the angle,\r\n" );
document.write( "    concluded by the given lines.\r\n" );
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document.write( "    From the other side, the center of the circle must lie on the perpendicular\r\n" );
document.write( "    to one of the given lines at the tangency point - so the center\r\n" );
document.write( "    of the circle is the intersection of the angle bisector and the \r\n" );
document.write( "    perpendicular to one of the tangency line at the tangency point.\r\n" );
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document.write( "What are the consequences from the fact that the problem is over-defined ?\r<BR>\n" );
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document.write( "The consequence is that when the center is found as the intersection point<BR>\n" );
document.write( "of two perpendiculars to the given lines at the tangency points,<BR>\n" );
document.write( "the person, who solves the problem, &nbsp;MUST &nbsp;check that the distance<BR>\n" );
document.write( "from the intersection point to the given tangency points &nbsp;IS &nbsp;THE &nbsp;SAME:\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It will guarantee that the condition of the problem <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;is self-consistent and is not self-contradictory.\r<BR>\n" );
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document.write( "Without such a check, &nbsp;the solution is formally incomplete; <BR>\n" );
document.write( "it is completed &nbsp;ONLY &nbsp;when the check is done.\r<BR>\n" );
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document.write( "Fortunately, &nbsp;in our case &nbsp;(it is easy to check) &nbsp;the distance from the intersection <BR>\n" );
document.write( "point &nbsp;(1,0) &nbsp;to the given tangency points is the same: &nbsp;it is equal to  &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "--------------------\r<BR>\n" );
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document.write( "<U>Comment from student</U> : &nbsp;&nbsp;It's easier of the author provided a graph.\r<BR>\n" );
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document.write( "<U>My response</U> : &nbsp;&nbsp;In &nbsp;Geometry, &nbsp;the plots are never considered as a proof <BR>\n" );
document.write( "or a tool to make a proof: &nbsp;the plots work and are used for visualization, &nbsp;ONLY.\r<BR>\n" );
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document.write( "Especially, &nbsp;in this problem, &nbsp;where the radius is an irrational number &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <BR>\n" );
document.write( "and you can not distinct visually &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285.1%29 ALIGN=MIDDLE  ALT=\"sqrt%285.1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So, &nbsp;your attempt to object or to argue my conception is invalid.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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