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Question Number 211127 by behi834171 last updated on 28/Aug/24

  Two circles are assumed that have the same center.   How many circles are there that are tangent to both circles and pass through the assumed point?

$$ \\ $$Two circles are assumed that have the same center. How many circles are there that are tangent to both circles and pass through the assumed point?

Commented by mr W last updated on 29/Aug/24

what do you mean with “the assumed  point”? where does this point lie?

$${what}\:{do}\:{you}\:{mean}\:{with}\:``{the}\:{assumed} \\ $$$${point}''?\:{where}\:{does}\:{this}\:{point}\:{lie}? \\ $$

Commented by mr W last updated on 29/Aug/24

if this point lies in the ring area,  then there are 4 circles which  tangent both circles and pass through  this point.

$${if}\:{this}\:{point}\:{lies}\:{in}\:{the}\:{ring}\:{area}, \\ $$$${then}\:{there}\:{are}\:\mathrm{4}\:{circles}\:{which} \\ $$$${tangent}\:{both}\:{circles}\:{and}\:{pass}\:{through} \\ $$$${this}\:{point}. \\ $$

Commented by mr W last updated on 29/Aug/24

Commented by behi834171 last updated on 29/Aug/24

  Thank you for your attention,sir mr.W.    The point can be anywhere: inside the any one of two concentric circles,or,on the side of the outer circle,or outside both circles.

$$ \\ $$Thank you for your attention,sir mr.W. The point can be anywhere: inside the any one of two concentric circles,or,on the side of the outer circle,or outside both circles.

Commented by mr W last updated on 29/Aug/24

only when the point lies between  the two concentric circles, as shown  in the example above, there exist  circles which tangent the both   circles.

$${only}\:{when}\:{the}\:{point}\:{lies}\:{between} \\ $$$${the}\:{two}\:{concentric}\:{circles},\:{as}\:{shown} \\ $$$${in}\:{the}\:{example}\:{above},\:{there}\:{exist} \\ $$$${circles}\:{which}\:{tangent}\:{the}\:{both}\: \\ $$$${circles}. \\ $$

Commented by behi834171 last updated on 29/Aug/24

Thank you so much dear master.  if the radii of cocentric circles  are given,  is it possible to find the radii of   tangent circles?

$${Thank}\:{you}\:{so}\:{much}\:{dear}\:{master}. \\ $$$${if}\:{the}\:{radii}\:{of}\:{cocentric}\:{circles}\:\:{are}\:{given}, \\ $$$${is}\:{it}\:{possible}\:{to}\:{find}\:{the}\:{radii}\:{of}\: \\ $$$${tangent}\:{circles}? \\ $$

Commented by mr W last updated on 29/Aug/24

yes. say the radius of big circle is R  and the radius of small circle is r,  then the radius of the circles which  tangent these two circles is  r_1 =((R−r)/2) (the red ones) or  r_2 =((R+r)/2) (the blues ones)

$${yes}.\:{say}\:{the}\:{radius}\:{of}\:{big}\:{circle}\:{is}\:{R} \\ $$$${and}\:{the}\:{radius}\:{of}\:{small}\:{circle}\:{is}\:{r}, \\ $$$${then}\:{the}\:{radius}\:{of}\:{the}\:{circles}\:{which} \\ $$$${tangent}\:{these}\:{two}\:{circles}\:{is} \\ $$$${r}_{\mathrm{1}} =\frac{{R}−{r}}{\mathrm{2}}\:\left({the}\:{red}\:{ones}\right)\:{or} \\ $$$${r}_{\mathrm{2}} =\frac{{R}+{r}}{\mathrm{2}}\:\left({the}\:{blues}\:{ones}\right) \\ $$

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