document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210573>Question 1210573</A>: <I><SPAN STYLE=\"word-break: break-all;\"> H is the orthocenter of acute triangle ABC and the extensions of AH, BH, and CH intersect the circumcircle of traingle ABC at A prime, B prime and C prime. We know angle AHB : angle BHC : angle CHA = 2 : 5 : 8. Find angle AprimeBprimeCprime in degrees. </SPAN>\n" );
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document.write( "H is the orthocenter of acute triangle ABC and the extensions of AH, BH, and CH intersect the circumcircle <BR>\n" );
document.write( "of triangle ABC at A', B' and C'. We know angle AHB : angle BHC : angle CHA = 2 : 5 : 8. Find angle A'B'C' in degrees.<BR>\n" );
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document.write( "This \"problem\" is SELF-CONTRADICTORY and describes a situation which NEVER may happen in Euclidian geometry.\r<BR>\n" );
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document.write( "Indeed, in an acute triangle, the orthocenter (the intersection point of three altitudes) is always inside<BR>\n" );
document.write( "the triangle. So, the angles of visibility to any side of an acute triangle from the orthocenter <BR>\n" );
document.write( "must be less than 180 degrees each.\r<BR>\n" );
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document.write( "Let's look at the given angles. We have 2 + 5 + 8 = 15 equal parts, so each part is  360/15 = 24 degrees.\r<BR>\n" );
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document.write( "Then angle AHB = 2*24 = 48 degrees,  angle BHC = 5*24 = 120 degrees and angle CHA = 8*24 = 192 degrees.\r<BR>\n" );
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document.write( "We see that angle CHA of 192 degrees is greater than 180 degrees, which, as we discussed above, may not happen.\r<BR>\n" );
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document.write( "So, the posed problem is a FAKE.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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