document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1113755>Question 1113755</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can a parallelogram always be inscribed in a sircle ? What about square,rhombus, kite and why . I also need reasons.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #728868 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=728868\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You know you can always inscribe a triangle in a circle.<BR>\n" );
document.write( "For any triangle,<BR>\n" );
document.write( "all perpendicular bisectors of the sides will intersect at the center of its circumcircle.<BR>\n" );
document.write( "To find the center of the circumcircle you only need two of those perpendicular bisectors.<BR>\n" );
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document.write( "A quadrilateral ABCD can be divided into two triangles by diagonal AC.<BR>\n" );
document.write( "You can draw the perpendicular bisectors of sides AB and BC.<BR>\n" );
document.write( "Where they intersect you find the center, O, of the one and only circumcircle for triangle ABC.<BR>\n" );
document.write( "Would point D also be on the circle?<BR>\n" );
document.write( "If it is, the perpendicular bisectors of sides AD and DC also have to go through point O.<BR>\n" );
document.write( "For a quadrilateral to be inscribed in a circle, the perpendicular bisectors of all four sides have to go through the center of the circle.<BR>\n" );
document.write( "The quadrilateral that can be inscribed in a circle is called a cyclical quadrilateral, or an inscribed quadrilateral.<BR>\n" );
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document.write( "For a rectangle ABCD, the perpendicular bisector of a pair of opposite sides<BR>\n" );
document.write( "is the same line connecting the midpoints of those sides. When you do that with both pairs of opposite sides, you just get just two lines that of course have a point in common><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28330%2C220%2C-3%2C3%2C-2%2C2%2C%0D%0Agreen%28rectangle%28-3%2C-2%2C3%2C0%29%29%2C%0D%0Agreen%28rectangle%280%2C-2%2C3%2C2%29%29%2C%0D%0Arectangle%28-3%2C-2%2C3%2C2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28rectangle%29 ALIGN=MIDDLE  ALT=\"highlight%28rectangle%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a cyclical quadrilateral, and can always be inscribed in a circle.<BR>\n" );
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document.write( "A <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28square%29 ALIGN=MIDDLE  ALT=\"highlight%28square%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a special kind of rectangle,so it can always be inscribed in a circle.<BR>\n" );
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document.write( "A rhombus is a special kind of parallelogram.<BR>\n" );
document.write( "A rhombus is also a special kind of kite, but a kite can only be a parallelogram if it is a rhombus.<BR>\n" );
document.write( "Here is a kite that is not a parallelogram:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28270%2C260%2C-1.8%2C3.6%2C-2.6%2C2.6%2C%0D%0Agreen%28triangle%28-1.6%2C0%2C3.4%2C0%2C0%2C2.4%29%29%2C%0D%0Agreen%28triangle%280%2C-2.4%2C0%2C2.4%2C3.4%2C0%29%29%2C%0D%0Agreen%28rectangle%280%2C0%2C0.2%2C0.2%29%29%2C%0D%0Aline%28-1.6%2C0%2C0%2C2.4%29%2Cline%280%2C2.4%2C3.4%2C0%29%2C%0D%0Aline%283.4%2C0%2C0%2C-2.4%29%2Cline%280%2C-2.4%2C-1.6%2C0%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > > you can see that the short sides are not parallel to the long sides.<BR>\n" );
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document.write( "A kite and a rhombus have diagonals that are perpendicular to each other, and one of those diagonals is an axis of symmetry.<BR>\n" );
document.write( "Some special kites can be inscribed in a circle,<BR>\n" );
document.write( "but not all kites can be inscribed in a circle.<BR>\n" );
document.write( "Let us figure out what would make them special that way.<BR>\n" );
document.write( "If the two diagonals have different lengths, consider the longest one.<BR>\n" );
document.write( "It is an axis of symmetry for the kite,<BR>\n" );
document.write( "so for that kite to be inscribed in a circle,<BR>\n" );
document.write( "that diagonal must be the diameter of the circle.<BR>\n" );
document.write( "In that case, each of the two triangles that diagonal divides the quadrilateral into<BR>\n" );
document.write( "is inscribed in the circle and has the circle diameter for its longest side. The inscribed angle opposite that side intercepts half a circle,<BR>\n" );
document.write( "so it is a right angle.<BR>\n" );
document.write( "For a kite to be a cyclical quadrilateral, it must have two right angles,<BR>\n" );
document.write( "which will be at the ends of a diameter that is not the longest diameter.<BR>\n" );
document.write( "Here is a kite that can be inscribed in a circle:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28270%2C270%2C-2.6%2C2.6%2C-2.6%2C2.6%2C%0D%0Agreen%28triangle%28-2%2C-1.5%2C-2%2C1.5%2C2%2C-1.5%29%29%2C%0D%0Agreen%28triangle%28-2%2C-1.5%2C2%2C-1.5%2C0.8%2C2.34%29%29%2C%0D%0Arectangle%28-2%2C-1.5%2C-1.8%2C-1.3%29%2C%0D%0Ared%28circle%280%2C0%2C2.5%29%29%2C%0D%0Aline%28-2%2C-1.5%2C2%2C-1.5%29%2Cline%28-2%2C-1.5%2C-2%2C1.5%29%2C%0D%0Aline%28-2%2C1.5%2C0.8%2C2.34%29%2Cline%280.8%2C2.34%2C2%2C-1.5%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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