document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=412610>Question 412610</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help:\r<BR>\n" );
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document.write( "Let a, b, c, d be four points on a ciricle. Let four more circles pass through a and b, b and c, c and d, d and a, respectively, meeting in further points a'b'c'd'. Show that a'b'c'd' is a cyclic quadrilateral\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 #290073 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=290073\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I was able to solve this problem entirely by angle chasing. However, drawing a diagram on this website is extremely difficult, so I'll try to be as verbose as possible (maybe another tutor will post a diagram). I recommend you draw a diagram on your paper, going step by step carefully.\r<BR>\n" );
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document.write( "Since ABCD are four points on a circle, then ABCD is cyclic. Suppose that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=omega%5B1%5D ALIGN=MIDDLE  ALT=\"omega%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the circle determined by A, B, A', B'.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=omega%5B2%5D ALIGN=MIDDLE  ALT=\"omega%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the circle determined by B, C, B', C'.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=omega%5B3%5D ALIGN=MIDDLE  ALT=\"omega%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the circle determined by C, D, C', D'.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=omega%5B4%5D ALIGN=MIDDLE  ALT=\"omega%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the circle determined by D, A, D', A'.\r<BR>\n" );
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document.write( "Suppose that angle ADD' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >, angle D'DC = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, angle CBB' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gamma ALIGN=MIDDLE  ALT=\"gamma\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and angle ABB' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since ABCD is cyclic, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha+%2B+beta+%2B+gamma+%2B+theta+=+180 ALIGN=MIDDLE  ALT=\"alpha+%2B+beta+%2B+gamma+%2B+theta+=+180\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Also, note that quadrilaterals ABB'A', BCC'B', CDD'C', and DAA'D' are all cyclic, as they are inscribed by circles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=omega%5B1%5D ALIGN=MIDDLE  ALT=\"omega%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ..., <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=omega%5B4%5D ALIGN=MIDDLE  ALT=\"omega%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
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document.write( "angle AA'D' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180+-+alpha ALIGN=MIDDLE  ALT=\"180+-+alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "angle D'C'C = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180+-+beta ALIGN=MIDDLE  ALT=\"180+-+beta\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "angle B'C'C = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180+-+gamma ALIGN=MIDDLE  ALT=\"180+-+gamma\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "angle B'A'A = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180+-+theta ALIGN=MIDDLE  ALT=\"180+-+theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, we can find that angle B'C'D' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360+-+%28180+-+beta%29+-+%28180+-+gamma%29+=+beta+%2B+gamma ALIGN=MIDDLE  ALT=\"360+-+%28180+-+beta%29+-+%28180+-+gamma%29+=+beta+%2B+gamma\"   DISABLED_style= \"max-width:90%; height:auto;\" > and angle B'A'D' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360+-+%28180+-+alpha%29+-+%28180+-+theta%29+=+alpha+%2B+theta ALIGN=MIDDLE  ALT=\"360+-+%28180+-+alpha%29+-+%28180+-+theta%29+=+alpha+%2B+theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha+%2B+beta+%2B+gamma+%2B+theta+=+180 ALIGN=MIDDLE  ALT=\"alpha+%2B+beta+%2B+gamma+%2B+theta+=+180\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then angle B'C'D' + angle B'A'D' = 180. They are opposite angles, so by definition, A'B'C'D' is cyclic. </SPAN>\n" );
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