document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195525>Question 1195525</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove the following theorem: There exists one circumcircle for any triangle, or any triangle is cyclic. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828038 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828038\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Consider any 3 non-collinear points  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "The center of the circle passing through any two given points lies on the perpendicular bisector of the line segment joining those points. This is easy to prove.\r<BR>\n" );
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document.write( "Let  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" >   be the center of any circle passing through the points  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >  . \r<BR>\n" );
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document.write( "Join <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+OA+ ALIGN=MIDDLE  ALT=\"+OA+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OB ALIGN=MIDDLE  ALT=\"OB\"   DISABLED_style= \"max-width:90%; height:auto;\" >   and draw  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OD+ ALIGN=MIDDLE  ALT=\"OD+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  be perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+AB ALIGN=MIDDLE  ALT=\"+AB\"   DISABLED_style= \"max-width:90%; height:auto;\" >  , touching <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+AB ALIGN=MIDDLE  ALT=\"+AB\"   DISABLED_style= \"max-width:90%; height:auto;\" >   at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+D ALIGN=MIDDLE  ALT=\"+D\"   DISABLED_style= \"max-width:90%; height:auto;\" >  .\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"https://ibb.co/FDmgZsb\"><IMG STYLE=\"max-width:80%;\" src=\"https://i.ibb.co/FDmgZsb/main-qimg-1002ab23e6b393154bc4873a6427a964.png\" alt=\"main-qimg-1002ab23e6b393154bc4873a6427a964\" border=\"0\"></a>\r<BR>\n" );
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document.write( "In  Δ  s <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAD ALIGN=MIDDLE  ALT=\"OAD\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OBD ALIGN=MIDDLE  ALT=\"OBD\"   DISABLED_style= \"max-width:90%; height:auto;\" > :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA=OB ALIGN=MIDDLE  ALT=\"OA=OB\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (radius) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OD ALIGN=MIDDLE  ALT=\"OD\"   DISABLED_style= \"max-width:90%; height:auto;\" >   is common<BR>\n" );
document.write( "∡<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ODA ALIGN=MIDDLE  ALT=\"ODA\"   DISABLED_style= \"max-width:90%; height:auto;\" > =∡<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ODB=90 ALIGN=MIDDLE  ALT=\"ODB=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∘  (By construction).\r<BR>\n" );
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document.write( "So,  Δ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAD ALIGN=MIDDLE  ALT=\"OAD\"   DISABLED_style= \"max-width:90%; height:auto;\" >=Δ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OBD ALIGN=MIDDLE  ALT=\"OBD\"   DISABLED_style= \"max-width:90%; height:auto;\" > . \r<BR>\n" );
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document.write( "Therefore  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=BD ALIGN=MIDDLE  ALT=\"AD=BD\"   DISABLED_style= \"max-width:90%; height:auto;\" > ⟹ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OD ALIGN=MIDDLE  ALT=\"OD\"   DISABLED_style= \"max-width:90%; height:auto;\" >⊥  bisector of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+AB+ ALIGN=MIDDLE  ALT=\"+AB+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now consider the perpendicular bisectors of line segments <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+AB ALIGN=MIDDLE  ALT=\"+AB\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > . \r<BR>\n" );
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document.write( "As the points  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+C ALIGN=MIDDLE  ALT=\"+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >  are not collinear, the perpendicular bisectors of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB+ ALIGN=MIDDLE  ALT=\"AB+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" >  are not parallel, and must intersect; two lines can intersect at only one point (let’s call it  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O+ ALIGN=MIDDLE  ALT=\"O+\"   DISABLED_style= \"max-width:90%; height:auto;\" >), which must then be the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=center+ ALIGN=MIDDLE  ALT=\"center+\"   DISABLED_style= \"max-width:90%; height:auto;\" >of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=circle ALIGN=MIDDLE  ALT=\"circle\"   DISABLED_style= \"max-width:90%; height:auto;\" > passing through the 3 given points (trivial to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA=OB=OC+ ALIGN=MIDDLE  ALT=\"OA=OB=OC+\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "As the points  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+C ALIGN=MIDDLE  ALT=\"+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >  are non-collinear, they could be the vertices of a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=triangle ALIGN=MIDDLE  ALT=\"triangle\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Hence the proof.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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