document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190325>Question 1190325</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A circle is inscribed in a square, which is inscribed in an equilateral triangle inscribed in another circle. What is the ratio of the area of the inner circle to the area of the outer circle? </SPAN>\n" );
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document.write( "Let the equilateral triangle be ABC with base AC and peak B. Let BD be the altitude of the triangle to side AC.  Let EF be the side of the square parallel to base AC but not on AC; EF intersects BD at G.<br><BR>\n" );
document.write( "Let r = OG be the radius of the inner circle.<br><BR>\n" );
document.write( "Triangle BGF is a 30-60-90 right triangle; and the length of GF is the radius of the circle, r; so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BG=r%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"BG=r%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "BO is the radius of the outer circle; and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BO=OG%2BBG=r%2Br%2Asqrt%283%29+=+r%281%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"BO=OG%2BBG=r%2Br%2Asqrt%283%29+=+r%281%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The radii of the inner and outer circle are then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%281%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"r%281%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the ratio of the areas of the inner and outer circles is<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F%281%2Bsqrt%283%29%29%29%5E2 ALIGN=MIDDLE  ALT=\"%281%2F%281%2Bsqrt%283%29%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Re-format as needed/desired....<br><BR>\n" );
document.write( " </SPAN>\n" );
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