document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=24523>Question 24523</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Somebody please help me with this problem... and thank you in advance!!\r<BR>\n" );
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document.write( "USING DEDUCTIVE REASONING<BR>\n" );
document.write( "There are 9 significant points associated with a triangle that always lie on a circle. These points are:<BR>\n" );
document.write( "- the midpoints of the sides: D, E, F<BR>\n" );
document.write( "- the feet of the altitudes: P, Q, R<BR>\n" );
document.write( "- the mipoints of the segments joining the orthocentre, H, to the three vertices: X, Y, Z\r<BR>\n" );
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document.write( "a) Prove that quadrilaterals FEZY and DEXY are rectangles<BR>\n" );
document.write( "b) Expain why the result of part a) proves that D, E, F, X, Y, and Z all lie on the same circle.<BR>\n" );
document.write( "c) Why do the points P, Q, and R also lie on this circle?<BR>\n" );
document.write( "d) Complete a proof that D, E, F, P, Q, R, X, Y, and Z lie on a circle. This circle is called the nin-point circle of the triangle. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #13125 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=13125\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> USING DEDUCTIVE REASONING<BR>\n" );
document.write( "There are 9 significant points associated with a triangle that always lie on a circle. These points are:<BR>\n" );
document.write( "- the midpoints of the sides: D, E, F<BR>\n" );
document.write( "- the feet of the altitudes: P, Q, R<BR>\n" );
document.write( "- the mipoints of the segments joining the orthocentre, H, to the three vertices: X, Y, Z\r<BR>\n" );
document.write( "\n" );
document.write( "a) Prove that quadrilaterals FEZY and DEXY are rectangles<BR>\n" );
document.write( "b) Expain why the result of part a) proves that D, E, F, X, Y, and Z all lie on the same circle.<BR>\n" );
document.write( "c) Why do the points P, Q, and R also lie on this circle?<BR>\n" );
document.write( "d) Complete a proof that D, E, F, P, Q, R, X, Y, and Z lie on a circle. This circle is called the nin-point circle of the triangle.<BR>\n" );
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document.write( "NOW THAT YOU HAVE NAMED ALL THE POINTS IN THE DIAGRAM,WE SHALL USE THOSE AND ASSUME YOU HAVE A DRAWING NAMED ACCORDINGLY IN FRONT OF YOU.<BR>\n" );
document.write( "a) Prove that quadrilaterals FEZY.WE HAVE <BR>\n" );
document.write( "IN TRIANGLE ABC,F AND E ARE MID POINTS OF AB AND AC.HENCE FE IS PARALLEL TO BC AND FE=BC/2 <BR>\n" );
document.write( "IN TRIANGLE HBC,Y AND Z ARE MID POINTS OF HB AND HC.HENCE YZ IS PARALLEL TO BC AND YZ=BC/2 <BR>\n" );
document.write( "SO FE=YZ.<BR>\n" );
document.write( "SIMILARLY<BR>\n" );
document.write( "IN TRIANGLE ABH,F AND Y ARE MID POINTS OF AB AND BH.HENCE FY IS PARALLEL TO AH AND FY=AH/2 <BR>\n" );
document.write( "IN TRIANGLE AHC,E AND Z ARE MID POINTS OF AC AND HC.HENCE EZ IS PARALLEL TO AH AND EZ=AH/2 <BR>\n" );
document.write( "SO FY=EZ.<BR>\n" );
document.write( "FURTHER AS WE PROVED ABOVE FY IS PARALLEL TO AH OR AHP AND YZ IS PARALLEL TO BC.BUT AHP IS PERPENDICULAR TO BC.HENCE FY IS PERPENDICULAR TO YZ.<BR>\n" );
document.write( "SIMILARLY WE CAN PROVE THAT FE IS PERPENDICULAR TO EZ.<BR>\n" );
document.write( "SO FEZY IS A RECTANGLE.<BR>\n" );
document.write( "SIMILARLY WE CAN PROVE THAT <BR>\n" );
document.write( "DEXY IS A RECTANGLE.<BR>\n" );
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document.write( "b) Expain why the result of part a) proves that D, E, F, X, Y, and Z all lie on the same circle.\r<BR>\n" );
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document.write( "WE PROVED ABOVE THAT <BR>\n" );
document.write( "quadrilaterals FEZY and DEXY are rectangles.FURTHER THEY HAVE A COMMON DIAGONAL EY.WE KNOW IN RECTANGLES DIAGONALS ARE EQUAL AND BISECT EACH OTHER.SO IN THESE 2 RECTANGLES DIAGONALS FZ,EY AND DX ARE ALL EQUAL AND BISECT EACH OTHE.LET THE POINT OF BISECTION BE N .SO IF WE DRAW A CIRCLE WITH N AS CENTRE AND ANY OF THE 3 DIAGONAL LENGTHS AS DIAMETER THEN ,SINCE ALL THE 3 DIAGONALS ARE EQUAL IN LEMGTH , THAT CIRCLE WILL PASS THROUGH  AL THE ENDS OF THE 3 DIAGONALS.HENCE F,E,Z,Y,D AND X ARE CONCYCLIC OR LIE ON THE SAME CIRCLE.<BR>\n" );
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document.write( "c) Why do the points P, Q, and R also lie on this circle?<BR>\n" );
document.write( "DX IS A DIAMETER OF THE ABOVE CIRCLE.WE KNOW THAT ANGLE DPX IS 90 DEGREES AS AP IS SAME AS BDPC IS PERPENDICULAR TO AP OR AXP.SINCE ANGLE IN A SEMICIRCLE IS 90 DEGREES . HENCE A CIRCLE WITH DX AS DIAMETER WILL PASS THROUGH P.<BR>\n" );
document.write( "SIMILARLY WE CAN PROVE THAT Q AND R ALSO LIE ON THE SAME CIRCLE <BR>\n" );
document.write( "d) Complete a proof that D, E, F, P, Q, R, X, Y, and Z lie on a circle. This circle is called the nin-point circle of the triangle.<BR>\n" );
document.write( "AS PROVED ABOVE ALL 9 POINTS  D, E, F, P, Q, R, X, Y, and Z lie on a circle\r<BR>\n" );
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document.write( " </SPAN>\n" );
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