document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195512>Question 1195512</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two equal chords AB and XY of a circle intersect at E, a point inside the circle. Use equal arcs to prove that AX=BY and that ∆EBX is isosceles. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828028 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52786)</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=828028\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Two equal chords AB and XY of a circle intersect at E, a point inside the circle. <BR>\n" );
document.write( "Use equal arcs to prove that <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;(a) &nbsp;&nbsp;AX=BY <BR>\n" );
document.write( "&nbsp;&nbsp;and that <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;(b) &nbsp;&nbsp;∆EBX is isosceles.<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Make a sketch on your own, as described in the problem.\r\n" );
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document.write( "So, you have a circle and two chords AB and XY in it.\r\n" );
document.write( "The chords intersect at the inner point E.\r\n" );
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document.write( "Since the chords are equal, the corresponding minor arcs AB and XY are congruent \r\n" );
document.write( "(have the same arc lengths).\r\n" );
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document.write( "Arc XB is the common part (the intersection) of minor arcs AB and XY.\r\n" );
document.write( "When you subtract (remove) this common part (the intercetion) from arcs AB and XY,\r\n" );
document.write( "you will get equal remaining arcs AX and BY. \r\n" );
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document.write( "So, arcs AX and BY have equal arc lengths.\r\n" );
document.write( "Due to the basic arcs property *), it implies than tighten chords AX and BY are congruent,\r\n" );
document.write( "i.e. have equal lengths: AX = BY.\r\n" );
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document.write( "QED.\r\n" );
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document.write( "First statement is proved.\r<BR>\n" );
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document.write( "*) &nbsp;&nbsp;Here this basic property is formulated explicitly:\r<BR>\n" );
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document.write( "    +---------------------------------------------+\r\n" );
document.write( "    |    Two chords in a circle are congruent     |\r\n" );
document.write( "    |             if and only if                  |\r\n" );
document.write( "    |    their corresponding arcs are congruent.  |\r\n" );
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document.write( "</PRE>\r<BR>\n" );
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document.write( "You may find it in any serious Geometry textbook. &nbsp;It is a part of the standard &nbsp;Geometry curriculum,<BR>\n" );
document.write( "which is a &nbsp;PRE-REQUISITE &nbsp;for solving such problems - so, &nbsp;I &nbsp;assume that you are familiar with it.\r<BR>\n" );
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document.write( "Otherwise, &nbsp;you may find the proof of this property in the lesson\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Circles/The-longer-is-the-chord-the-larger-its-central-angle-is.lesson>The longer is the chord the larger its central angle is</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Next, &nbsp;the proof of the second statement is in couple of lines.\r<BR>\n" );
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document.write( "In triangle EXB, consider angles EXB and EBX.\r\n" );
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document.write( "These angles lean on arcs BY and AX, respectively.\r\n" );
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document.write( "But these arcs are congruent, as I proved it above.\r\n" );
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document.write( "Hence, angles EXB and EBX are congruent.\r\n" );
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document.write( "Thus, triangle EXB is isosceles, since it has congruent base angles at tyhe bas XB.\r\n" );
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document.write( "The proof and the solution is complete.\r<BR>\n" );
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document.write( "At this site, &nbsp;you have free of charge online textbook on Geometry\r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "The referred lesson is the part of this online textbook under the topic &nbsp;\"<U>Properties of circles, their chords, secants and tangents</U>\".\r<BR>\n" );
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document.write( "Save the link to this online textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in GEOMETRY<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson\r<BR>\n" );
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document.write( "to your archive and use it when it is needed.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This textbook contains &nbsp;whole/(entire) &nbsp;standard high-school \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Geometry curriculum in its logical development from &nbsp;\" a \" &nbsp;to &nbsp;\" Z \".\r<BR>\n" );
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document.write( " </SPAN>\n" );
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