三角形五心及其性质

ABC ADBECFDEFHABC abcp=(a+b+c)/212、三角形的垂心是它垂足三角形的内心；或者说，H ABCABC ( )AH・HD=BH・HE=CH・HFHABC称这样的四点为一—垂心组)。ABCABHBCHACHH ABACPQ AB/AP・tanB+AC/AQ・tanC=tanA+tanB+tanC9 OH ABCBAO=HACABH=OBCBCO=HCA10211接三角形（顶点在原三角形的边上）中，1213ABC T么T是垂PB*PC*BC+PB*PA*AB+PA*PC*AC=AB*BC*CAH ABC OH=hOA=aOB=bOC=ch=(tanAa+tanBb+tanCc)/(tanA+tanB+tanC).(a1 b1)(a2b2)(a3b3)x=x/2/ y=-y/2/=det([x2-x1 x3-x2y2-y1 y3-y2]);X=det([(x1+x2)*(x2-x1)+(y1+y2)*(y2-y1)y2-y1;(x2+x3)*(x3-x2)+(y2+y3)*(y3-y2)y3-y2]);y=det([x3-x2(y2+y3)*(y3-y2);x3-x1(y3+y1)*(y3-y1)+(x2-x1)*(x1-x3)]);ABCOA・OB=OB・OC=OC・OAOOA・OB=OB・OCOA・OB-OC・OB=0(OA-OC)・OB=0CA・OB=0 OB ACOB・OC=OC・OA OC ABOA・OB=OC・OA OA BCABCDBCEACADBEOCOABFFABSAOB=SAOCSAOB=SBOCSAOC=SBOCAF=BF1.212.333.34.((X1+X2+X3)/3,(Y1+Y2+Y3)/3)(X1+X2+X3)/3 (Y1+Y2+Y3)/3Z1+Z2+Z3/336.ABCaOA=bOB+cOCOABCaOA=bOB+cOC111 12 13 p(I1/p,l2/p,l3/p)/ABC 00(r)Aabcp=(a+b+c)/2rr=S/pSABC=SOAB+SOAC+S0BC=(cr+br+ar)/2=rp,ABCC=90 r=(a+b-c)/2BOC=90+A/2O ABC O /ABC

a(OA)+b(OB)+c(OC)=07OABCI/ABCOl=[a(OA)+b(OB)+c(OC)]/(a+b+c)8/ABCA(x1y1)B(x2y2)C(x3y3) /ABCI(ax1/(a+b+c)+bx2/(a+b+c)+cx3/(a+b+c)ay1/(a+b+c)+by2/(a+b+c)+cy3/(a+b+c)( )/ABCRrOI OIA2=RA2-2Rr10ABCADADBCabcABCADADBCabcABC，d=ADb/sinB=c/sinCABC，d=ADb/sinB=c/sinCd=R1sinB=R2sinCR1ABDR2ACDd=R1sinB=R2sinCR1ABDR2ACDR1/R2=sinC/sinB=c/b.BD=R1sinBADCD=R2sinCAD,CAD=BADBD/CD=R1/R2=c/b=AB/ACCD=R2sinCAD,CAD=BADBD/CD=R1/R2=c/b=AB/AC/ABC OG(R)ABC abcp=(a+b+c)/21123 .2BGC=2ABGC=2(180-A).3G ABCG/ABC(GA+GB)AB=(GB+ GC)BC=(GC+GA)CA=0.4GABCPABCG/ABC1 PG=((tanB+tanC)PA+(tanC+tanA)PB+(tanA+tanB)PC)/2(tanA+tanB+tanC).2PG=(cosA/2sinBsin