document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252016>Question 252016</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q.2:- Let ABC be a triangle in which AB=AC and let I be its in-centre. suppose BC=AB+AI.find the angle BAC? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #484268 by </B><A HREF=/tutors/aboutme.mpl?userid=plastery><B>plastery(3)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=plastery><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=484268\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The angle BAC that satisfies the relation BC=AB+AI is the right angle.<BR>\n" );
document.write( "To get to this solution with some trigonometry, let's call x the angle BAC and l the lenght of AB..<BR>\n" );
document.write( "As the triangle ABC is isosceles and the sum of the internal angles of any triangle is 2 right angles, then the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC=ACB=%28PI-x%29%2F2=PI%2F2+-X%2F2 ALIGN=MIDDLE  ALT=\"ABC=ACB=%28PI-x%29%2F2=PI%2F2+-X%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Let me remind that the Incenter of a triangle can be found as the intersection of any two internal angle bisectors.<BR>\n" );
document.write( "Let's call H the intersection of the bisector of BAC and BC.<BR>\n" );
document.write( "AH is also the altitude relative to the base BC. So ABH is a right triangle (right in H).<BR>\n" );
document.write( "For the definition of sine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BH=BA%2Asin%28x%2F2%29=l+%2Asin%28x%2F2%29 ALIGN=MIDDLE  ALT=\"BH=BA%2Asin%28x%2F2%29=l+%2Asin%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the  Pythagorean theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AH=sqrt%28l%5E2-l%5E2%2Asin%5E2%28x%2F2%29%29=l%2Asqrt%281-sin%5E2%28x%2F2%29%29=l%2Asqrt%28cos%5E2%28x%2F2%29%29=l%2Acos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The angle IBH is half of ABH that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=IBH=ABC+%2F+2=+PI%2F4-x%2F4 ALIGN=MIDDLE  ALT=\"IBH=ABC+%2F+2=+PI%2F4-x%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the definition of tangent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=IH=BH%2Atan%28IBH%29=l%2Asin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29 ALIGN=MIDDLE  ALT=\"IH=BH%2Atan%28IBH%29=l%2Asin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finally <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AI=AH-IH=l%2Acos%28x%2F2%29-l%2Asin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29 ALIGN=MIDDLE  ALT=\"AI=AH-IH=l%2Acos%28x%2F2%29-l%2Asin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore, as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=2%2ABH ALIGN=MIDDLE  ALT=\"BC=2%2ABH\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the relation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=AB%2BAI ALIGN=MIDDLE  ALT=\"BC=AB%2BAI\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be written <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Al+%2Asin%28x%2F2%29=l%2Bl%2Acos%28x%2F2%29-l%2Asin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29 ALIGN=MIDDLE  ALT=\"2%2Al+%2Asin%28x%2F2%29=l%2Bl%2Acos%28x%2F2%29-l%2Asin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and with some trivial passage<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%28x%2F2%29=%281%2Bcos%28x%2F2%29-sin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29%29 ALIGN=MIDDLE  ALT=\"2%2Asin%28x%2F2%29=%281%2Bcos%28x%2F2%29-sin%28x%2F2%29%2Atan%28PI%2F4-x%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Still Algebra:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282%2Btan%28PI%2F4-x%2F4%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE  ALT=\"sin%28x%2F2%29%2A%282%2Btan%28PI%2F4-x%2F4%29%29=1%2Bcos%28x%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we apply the sum identity of the tangent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28alpha-beta%29=%28tan%28alpha%29-tan%28beta%29%29%2F%281%2Btan%28alpha%29%2Atan%28beta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > to obtain:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282%2B%28tan%28PI%2F4%29-tan%28x%2F4%29%29%2F%281%2Btan%28PI%2F4%29%2Atan%28x%2F4%29%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Known that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28PI%2F4%29=1 ALIGN=MIDDLE  ALT=\"tan%28PI%2F4%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we obtain:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282%2B%281-tan%28x%2F4%29%29%2F%281%2Btan%28x%2F4%29%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now let's apply the half angle formula for the tangent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28alpha%2F2%29=%281-cos%28alpha%29%29%2Fsin%28alpha%29 ALIGN=MIDDLE  ALT=\"tan%28alpha%2F2%29=%281-cos%28alpha%29%29%2Fsin%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282%2B%281-%281-cos%28x%2F2%29%29%2Fsin%28x%2F2%29%29%2F%281%2B%281-cos%28x%2F2%29%29%2Fsin%28x%2F2%29%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again Algebra and provided that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%3C%3E0 ALIGN=MIDDLE  ALT=\"sin%28x%2F2%29%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > i.e <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E0 ALIGN=MIDDLE  ALT=\"x%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we obtain:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282%2B%28sin%28x%2F2%29-1%2Bcos%28x%2F2%29%29%2F%28sin%28x%2F2%29%2B1-cos%28x%2F2%29%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282-%281-sin%28x%2F2%29-cos%28x%2F2%29%29%2F%281%2Bsin%28x%2F2%29-cos%28x%2F2%29%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%282%2B2%2Asin%28x%2F2%29-2%2Acos%28x%2F2%29-1%2Bsin%28x%2F2%29%2Bcos%28x%2F2%29%29%2F%281%2Bsin%28x%2F2%29-cos%28x%2F2%29%29%29=1%2Bcos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and provided that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%2F2%29-cos%28x%2F2%29%29%3C%3E0 ALIGN=MIDDLE  ALT=\"%281%2Bsin%28x%2F2%29-cos%28x%2F2%29%29%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2A%281%2B3%2Asin%28x%2F2%29-cos%28x%2F2%29%29=%281%2Bcos%28x%2F2%29%29%2A%281%2Bsin%28x%2F2%29-cos%28x%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29%2B3%2Asin%5E2%28x%2F2%29-sin%28x%2F2%29%2Acos%28x%2F2%29%29=1%2Bcos%28x%2F2%29%2Bsin%28x%2F2%29%2Bsin%28x%2F2%29%2Acos%28x%2F2%29-cos%28x%2F2%29-cos%5E2%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asin%5E2%28x%2F2%29-sin%28x%2F2%29%2Acos%28x%2F2%29%29=1-cos%5E2%28x%2F2%29%2Bsin%28x%2F2%29%2Acos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asin%5E2%28x%2F2%29-sin%28x%2F2%29%2Acos%28x%2F2%29%29=sin%5E2%28x%2F2%29%2Bsin%28x%2F2%29%2Acos%28x%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%5E2%28x%2F2%29-2%2Asin%28x%2F2%29%2Acos%28x%2F2%29%29=0 ALIGN=MIDDLE  ALT=\"2%2Asin%5E2%28x%2F2%29-2%2Asin%28x%2F2%29%2Acos%28x%2F2%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%28x%2F2%29%2A%28sin%28x%2F2%29-cos%28x%2F2%29%29=0 ALIGN=MIDDLE  ALT=\"2%2Asin%28x%2F2%29%2A%28sin%28x%2F2%29-cos%28x%2F2%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The first term cannot be zero for the previous condition, so the possible solution must satisfy:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2F2%29-cos%28x%2F2%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%2F2%29-cos%28x%2F2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "that happens when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2=PI%2F4 ALIGN=MIDDLE  ALT=\"x%2F2=PI%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > i.e. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=PI%2F2 ALIGN=MIDDLE  ALT=\"x=PI%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The general solution would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%2API%2An-3%2F2%2API ALIGN=MIDDLE  ALT=\"x=2%2API%2An-3%2F2%2API\"   DISABLED_style= \"max-width:90%; height:auto;\" > with n any integer, but for the geometric nature of the problem we can rest on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "Eventually we check the solution that is that in case of a right/isosceles triangle the relation BC=AB+AI is satisfied.<BR>\n" );
document.write( "With the same notation above:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=sqrt%282%2Al%5E2%29=l%2Asqrt%282%29=l%2A%281%2B%28sqrt%282%29-1%29%29 ALIGN=MIDDLE  ALT=\"BC=sqrt%282%2Al%5E2%29=l%2Asqrt%282%29=l%2A%281%2B%28sqrt%282%29-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "BH is such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=l%5E2=2%2ABH%5E2 ALIGN=MIDDLE  ALT=\"l%5E2=2%2ABH%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BH=l%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"BH=l%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC=PI%2F2 ALIGN=MIDDLE  ALT=\"ABC=PI%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABH=PI%2F4 ALIGN=MIDDLE  ALT=\"ABH=PI%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=IBH=PI%2F8 ALIGN=MIDDLE  ALT=\"IBH=PI%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=IH=BH%2Atan%28PI%2F8%29=BH%2A%281-cos%28PI%2F4%29%29%2Fsin%28Pi%2F4%29=BH%2A%281-sqrt%282%29%2F2%29%2F%28sqrt%282%29%2F2%29=BH%2A%282-sqrt%282%29%29%2Fsqrt%282%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AH=BH=l%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"AH=BH=l%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AI=AH-IH=l%2Fsqrt%282%29-l%2Fsqrt%282%29%2A%282-sqrt%282%29%29%2Fsqrt%282%29=l%2A%28sqrt%282%29-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB%2BAI=l%2Bl%2A%28sqrt%282%29-1%29=l%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"AB%2BAI=l%2Bl%2A%28sqrt%282%29-1%29=l%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=2%2ABH=2%2Al%2Fsqrt%282%29=l%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"BC=2%2ABH=2%2Al%2Fsqrt%282%29=l%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Quod Demonstrandum Erat </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );