Let AB be divided internally and externally at R and S in the same ratio. Then AR, AB, AS are in

AB¯ is divided  by R&S internally & Externally in the same ratioLet R, S divide AB¯ harmonically in the ratio m:n⇒  ARRB=mn=ASSB  ⇒  RBAR=nm=SBAS⇒  RBAR+1=nm+1  ⇒  RB+ARAR=n+mm1−SBAS=1−nmAS−SBAS=m−nm  ⇒  ABAR=m+nm, ABAS=m−nmABAR+ABAS=2mm  ⇒  1AR+1AS=2AB⇒  AR, AB, AS  are in H.P.