document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=565708>Question 565708</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two circles meet at points P and Q, and diameters P A and P B<BR>\n" );
document.write( "are drawn. Show that the line AB goes through the point Q. (Probably it is easier to<BR>\n" );
document.write( "think of drawing the lines AQ and QB and then showing that they are actually the<BR>\n" );
document.write( "same line.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #366017 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=366017\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C270%2C0%2C10%2C0%2C9%2C%0D%0A%0D%0A%0D%0Acircle%284%2C4%2C3%29%2C%0D%0Acircle%287%2C4%2C2.3%29%2C%0D%0Alocate%286%2C7%2CP%29%2C%0D%0Alocate%286.5%2C2%2CQ%29%2C%0D%0Aline%286.1183%2C6.1243%2C+8%2C2%29%2C%0D%0Aline%286.1183%2C6.1243%2C+1.9%2C1.9%29%2C%0D%0Alocate%288%2C2%2CB%29%2C%0D%0Alocate%281.9%2C1.9%2CA%29%2C%0D%0Aline%281.9%2C1.9%2C8%2C2%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The easiest solution is probably to draw the segment connecting the centers of the circles (denote Y,Z), as well as segment PQ:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C270%2C0%2C10%2C0%2C9%2C%0D%0A%0D%0A%0D%0Acircle%284%2C4%2C3%29%2C%0D%0Acircle%287%2C4%2C2.3%29%2C%0D%0Alocate%286%2C7%2CP%29%2C%0D%0Alocate%286.5%2C2%2CQ%29%2C%0D%0Aline%286.1183%2C6.1243%2C+8%2C2%29%2C%0D%0Aline%286.1183%2C6.1243%2C+1.9%2C1.9%29%2C%0D%0Alocate%288%2C2%2CB%29%2C%0D%0Alocate%281.9%2C1.9%2CA%29%2C%0D%0Aline%281.9%2C1.9%2C8%2C2%29%2C%0D%0Aline%284%2C4%2C7.05%2C4.05%29%2C%0D%0Aline%286.1183%2C6.1243%2C6.2%2C2%29%2C%0D%0Alocate%284%2C4%2CY%29%2C%0D%0Alocate%287.05%2C4.05%2CZ%29%2C%0D%0Alocate%286.3%2C3.8%2CR%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since PY = (1/2)PA and PZ = (1/2)PB, triangles YPZ and APB are similar with a 1:2 ratio. Additionally, PR = (1/2)PQ (this can be proven by symmetry). Since R lies on YZ, Q must lie on AB.\r<BR>\n" );
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document.write( "Or, another way you can prove it is show that the pairs of triangles PRY/PQA and PRZ/PQB are similar. Then, you may let angle PRY = m, angle PQA = m, it follows that angle PRZ = angle PQB = 180-m. Hence, angles PQA + PQB = 180, so A,Q,B are collinear. </SPAN>\n" );
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