document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1031420>Question 1031420</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello I want to prove this T.T but i dont know how I need to draw this diagram and prove it!\r<BR>\n" );
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document.write( "Two circles intersect at points A and B. From any point P on the line AB, tangents PQ and PR are drawn to the circles. Prove that PQ=PR.\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #646263 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=646263\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Hello I want to prove this T.T but i dont know how I need to draw this diagram and prove it!\r<BR>\n" );
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document.write( "Two circles intersect at points A and B. From any point P on the line AB, tangents PQ and PR <BR>\n" );
document.write( "are drawn to the circles. Prove that PQ=PR.<BR>\n" );
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document.write( "We are given two circles with the centers O1 and O2 intersec-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<BR>\n" );
document.write( "ting in the common points A and B (see the Figure on the right).\r<BR>\n" );
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document.write( "The straight line AB is drawn through the points A and B.\r<BR>\n" );
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document.write( "The point P is an arbitrary point in this straight line.\r<BR>\n" );
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document.write( "The tangent lines PQ and PR are drawn from P to the circles O1 <BR>\n" );
document.write( "and O2 respectively. Q and P are the tangent points.\r<BR>\n" );
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document.write( "We need to prove that the tangent segments PQ and PR are <BR>\n" );
document.write( "congruent: |PQ| = |PR|.\r<BR>\n" );
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document.write( "The key for the proof is this Theorem of the school Geometry:\r<BR>\n" );
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document.write( "If a tangent and a secant lines are released from a point <BR>\n" );
document.write( "outside a circle, &nbsp;then the product of the measures <BR>\n" );
document.write( "of the secant and its external part is equal to the square <BR>\n" );
document.write( "of the tangent segment.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+400%2C+-3.5%2C+10.5%2C+-5.5%2C+10.5%2C+%0D%0A++++++++++++circle%280.0%2C+0.0%2C+3%2C+3%29%2C%0D%0A%0D%0A++++++++++++circle%280.0%2C+0.0%2C+0.05%2C+0.05%29%2C%0D%0A++++++++++++locate+%28-0.1%2C+-0.1%2C+O1%29%2C%0D%0A%0D%0A++++++++++++circle%285.0%2C+0.0%2C+4%2C+4%29%2C%0D%0A%0D%0A++++++++++++circle%285.0%2C+0.0%2C+0.05%2C+0.05%29%2C%0D%0A++++++++++++locate+%284.9%2C+-0.1%2C+O2%29%2C%0D%0A%0D%0A++++++++++++locate+%282%2C++3.0%2C+A%29%2C%0D%0A++++++++++++locate+%282%2C+-2.5%2C+B%29%2C%0D%0A%0D%0A++++++++red%28line%281.8%2C+-3%2C+1.8%2C+10%29%29%2C%0D%0A++++++++++++circle%281.8%2C+9%2C+0.08%2C+0.08%29%2C%0D%0A++++++++++++locate%282%2C+9.2%2C+P%29%2C%0D%0A%0D%0A+++++++blue%28line%281.8%2C+9%2C+-3.65%2C+0%29%29%2C%0D%0A++++++++++++circle%28-2.5%2C+1.9%2C+0.05%2C+0.05%29%2C%0D%0A++++++++++++locate%28-2.9%2C+2.3%2C+Q%29%2C%0D%0A%0D%0A+++++++blue%28line%281.8%2C+9%2C+11.2%2C+0%29%29%2C%0D%0A++++++++++++circle%287.6%2C+3.5%2C+0.05%2C+0.05%29%2C%0D%0A++++++++++++locate%287.7%2C+3.9%2C+R%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Regarding this theorem, see the lesson <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Circles/Metric-relations-for-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Metric relations for a tangent and a secant lines released from a point outside a circle</A> <BR>\n" );
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document.write( "Now, by applying this Theorem, we have &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28PQ%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28PQ%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = |PA|*|PB|,  &nbsp;&nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28PR%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28PR%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = |PA|*|PB|.\r<BR>\n" );
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document.write( "Since the right sides are identical, the left sides are equal: &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28PQ%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28PQ%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28PR%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28PR%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Then &nbsp;&nbsp;|PQ| = |PR|, &nbsp;&nbsp;QED.\r<BR>\n" );
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