document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158992>Question 1158992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> All triangles and rectangles have circumscribed circles. Is this true for all kites, trape-<BR>\n" );
document.write( "zoids, and parallelograms? Which quadrilaterals have circumscribed circles? Explain. </SPAN>\n" );
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document.write( "Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a con-cyclic polygon because its vertices are con-cyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic.\r<BR>\n" );
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document.write( "kites: \r<BR>\n" );
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document.write( "all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=right ALIGN=MIDDLE  ALT=\"right\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+kites ALIGN=MIDDLE  ALT=\"+kites\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cyclic ALIGN=MIDDLE  ALT=\"+cyclic\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the right kite is a convex quadrilateral and has two opposite right angles<BR>\n" );
document.write( "all right kites are<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+bicentric ALIGN=MIDDLE  ALT=\"+bicentric\"   DISABLED_style= \"max-width:90%; height:auto;\" > quadrilaterals (quadrilaterals with both a circumcircle and an in-circle)\r<BR>\n" );
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document.write( "trapezoid:\r<BR>\n" );
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document.write( "if the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=trapezoid ALIGN=MIDDLE  ALT=\"trapezoid\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the diagonals are of equal length, the two pairs of diagonally opposite angles will have equal sums, this sum being 180 degrees as the four angles must add to 360 degrees <BR>\n" );
document.write( "this means that an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+trapezoid+ ALIGN=MIDDLE  ALT=\"+trapezoid+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cyclic ALIGN=MIDDLE  ALT=\"cyclic\"   DISABLED_style= \"max-width:90%; height:auto;\" > quadrilateral, and thus by definition can be circumscribed by a circle \r<BR>\n" );
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document.write( "parallelograms:\r<BR>\n" );
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document.write( "any parallelogram cannot be cyclic<BR>\n" );
document.write( "for any quadrilateral to be cyclic the sum of the opposite angles should be 180 deg, so the parallelograms that can be cyclic are definitely a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=square+ ALIGN=MIDDLE  ALT=\"square+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rectangle ALIGN=MIDDLE  ALT=\"rectangle\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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