Equation of first circle x2+y2+2gx+2fy=0 can be written as (x+g)2+(y+f)2=g2+f2 Center of first circle (−g,−f) and radius √g2+f2 Equation of second circle x2+y2+2g′s+2f′y=0 can be written as (x+g′)2+(y+f′)2=g′2+f′2 Center of first circle (−g′,−f′) and radius √g′2+f′2 The two circle touch each other so √(−g+g′)2+(−f+f′)2=√g2+f2−√g′2+f′2 By squaring on both sides we get (−g+g′)2+(−f+f′)2=(√g2+f2−√g′2+f′2)2 gg′+ff′=√g2+f2√g′2+f′2 Again by squaring on both sides we get (gg′+ff′)2=(√g2+f2√g′2+f′2)2 2gg′ff′=(gf′2)+(g′f)2 (gf′2)+(g′f)2−2gg′ff′=0 (gf′−g′f)2=0⇒gf′−g′f=0 gf′=g 'f