2Q-yy ca(pmyylJ<y(R y 1C yy(;y(5D# D9:\x\()r3rZProof that the altitudes of any triangle are concurrent: Identity: If segments AD, BE, and CF are concurrent, then AF/FD * BD/DC * CE/EA = 1. Given that GH is parallel to CB, several triangles are similar: AHF to BCF, AEG to BCE, AGK to BDK, and CDK to AHK. By the definition of similar triangles, then, several proportions can be constructed: 1. AF/FB = AH/BC, (*) 2. CE/EA = BC/AG, (*) 3. AG/BD = AK/DK, and 4. AH/DC = AK/DK. From 3 and 4, it can be concluded that AG/BD = AH/DC. Thus, BD/DC = AG/AH (*). Multiplying all of the proportions marked by *, you get the product: AF/FB * BD/DC * CE/EA = AH/BC * BC/AG * AG/AH = (AH * BC * AG)/(BC * AG * AH) = 1. Therefore, if AD, BE, and CF intersect at K, the identity holds. Now we must prove that if two altitudes are concurrent, so is the third: Assume K is the point of intersection of BE and CF and draw line AK until it intersects with BC at D'. From the proven part of the theorem, it can be inferred that AF/FB * BD'/D'C * CE/EA = 1. On the other hand, it is given that AF/FB * BD/DC * CE/EA = 1. Combining these two statements, we get that: BD'/D'C = BD/DC or BD'/D'C + 1 = BD/DC + 1 or (BD' + D'C)/D'C = (BD + DC)/DC. Finally, BC/D'C = BC/DC which obviously implies that D'C = DC. That is to say D' and D are the same point. Therefore, AD, BE, and CF are concurrent.` G P %R%jC  /d yy(;yArp~8"tLP m8'<OG:af\E{[DCA9Ff yy(;ym3===^^D % m{!:A}BAC = eиy(!"5eKy(Hy("yy5ǐ&t"&&,y"yǐ&t"&$,"&,'T"y4xy(y@BH  \y@4 ywp($ Angle(BAC) = 5y"H4lh@yH (B*Dǐ&t" 3yyD &P@@3y0yy> ,t@@y h@yE"-(y y %,y(.Uy"(X y yzy(EyD @@y@"@yPy`yy "7~ y~7  ,7yy"U~5 "0~>y"yXy\y`(yypyy(y(~pyp$1f yy(;ym2D % m{!:A}CBA = eиy(!"5eKy(Hy("yy5ǐ&t"&&,y"yǐ&t"&$,"&,'T"y4xy(y@BH  \y@4 ywp($ Angle(CBA) = 5y"H4lh@yH (B*Dǐ&t" 3yyD &P@@3y0yy> ,t@@y h@yE"-(y y %,y(.Uy"(X y yzy(EyD @@y@"@yPy`yy "7~ y~7  ,7yy"U~5 "0~>y"yXy\y`(yypyy(y(~pypK Xe yy(;ym1D % m{!:A}ACB = eиy(!"5eKy(Hy("yy5ǐ&t"&&,y"yǐ&t"&$,"&,'T"y4xy(y@BH  \y@4 ywp($ Angle(ACB) = 5y"H4lh@yH (B*Dǐ&t" 3yyD &P@@3y0yy> ,t@@y h@yE"-(y y %,y(.Uy"(X y yzy(EyD @@y@"@yPy`yy "7~ y~7  ,7yy"U~5 "0~>y"yXy\y`(yypyy(y(~pyped  yy(;ylsnR;*sM 6B  DCC CA?eB  yy(;ykn&tT"rp^7LC CC ? 0$  yy(;yj (1:&       - 0DCAC>C ?/ yy(;yx(DCBCCB?  yy(; yuDCC CC ? S1lZlZ yy(; ytlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZllZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZDCBCX?[lZlZ yy(; yslZlZlZlZlZlZlZlZlZlZlZlZlZlZlZllZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZlZDCCbB? [s yy(;yoDC̀C-CB?  Y yy(;ynATUW>)giN ~ ^ DClCB?  yy(;ymT- jn$],w4:Y\=X8cLDCC7CwB?EKCN yy(;yw(aI6~6~QBDClCBCB?  yy(;yv(BI6~6~QBC:\WINDOWS\SYSTEM\TESCKEY.DC4ACA?   yy(;yD(# #$  p~ʢDCC  7<$ yy(;yF # #  pʢDCBce =B yy(;yE # #  p~ʢDClíB{d  yy(;yHDCCz    yy(;yrDCC CA?ھ 6$  yy(;yqDCBceC>C ?I<e  yy(;ypDClíB{dCC ?VU MR  yy(;yKh # #d  pʢDCB[s yy(;yz,]]DC2%Az,CB? yy(;yyJC:\SKETCH\gskethp.exeDC|A1XCoBr?IN yy(;yH8 # #4  p~ʢDCeA  yy(;yG # #  prʢDC-A