The number of circles touching all the three given lines $2x-5y+4=0,$  $3x+7y-5=0,$and $6x-15y+7=0$ is

If the lines form a triangle, then the number of circles which touches all three lines is 4

If the lines do not form a triangle, and two of them are parallel, third one is not parallel then the number circles which touches three lines is 2

If the lines do not form a triangle, and three are parallel lines, then the number of circles which touches all the lines is zero. 

If the lines doe not forms a triangle, and three are concurrent, then the number of circles which touches three lines is one.

Given lines are 

$\begin{array}{r}2x-5y+4=0----\left(1\right)\\ 3x+7y-5=0----\left(2\right)\\ 6x-15y+7=0----\left(3\right)\end{array}$

$\text{ The slope of the line }cx+by+c=0\text{ is }-\frac{a}{b}$

$\text{ Observing the equations }\left(1\right)\text{ and }\left(3\right),\text{ their slopes are }{m}_1=\frac{2}{5},m_3=\frac{2}{5}\text{, }$

Since the slopes are equal, those two lines are parallel, and the other line is not parallel to these two, the number of circles which touches all three lines is 2