Two circles having equal radii #r_1# and touching a line #l#on the same side of #l# are at a distance of #x# from each other. Third circle of radius #r_2# touches the two circles. How do we find the height of third circle from #l#?

Question

Wyatt Anderson · Accepted Answer

See below.

Supposing that #x# is the distance between perimeters and supposing that #2(r_1+r_2)gt x+2r_1# we have #h = sqrt((r_1+r_2)^2-(r_1+x/2)^2)+r_1-r_2# #h# is the distance between #l# and the perimeter of #C_2#

Jayden Jackson · Answer

undefined To find the height of the third circle from line $l$, let's assume:

- The radii of the two equal circles are $r_1$.
- The radius of the third circle is $r_2$.
- The distance between the centers of the two equal circles is $x$.

Given that the two circles of radius $r_1$ touch line $l$ on the same side and are $x$ units apart, their centers are $r_1$ units above $l$, and the distance $x$ is measured center to center.

The third circle of radius $r_2$ touches both of these circles. The key to solving this problem lies in understanding that when three circles touch each other externally, the centers of the circles form a triangle. In this case, we have a triangle where two sides are $r_1 + r_2$ (the sum of radii of the third circle and one of the two equal circles, forming two sides of the triangle), and the third side is $x$, the distance between the centers of the two equal circles.

The height of the third circle from line $l$ can be found by adding $r_1$, the height of the equal circles' centers from $l$, to the perpendicular distance from the third circle's center to line $l$, which goes through the triangle's incenter (the point where the angle bisectors of a triangle intersect, which is also the center of the circle inscribed in the triangle).

The formula to find the distance $d$ from the incenter to a side of the triangle (in our case, the distance from the third circle's center to line $l$) is given by the area of the triangle divided by the semiperimeter of the triangle. However, calculating this directly involves knowing $x$ and being able to calculate the triangle's area and semiperimeter.

A simpler approach is to visualize or calculate that the height of the third circle from line $l$ is $r_1 + r_2$, considering that the third circle touches both of the equal circles externally and the line $l$. This is because the center of the third circle is vertically aligned with the points where it touches the other two circles, forming a right triangle with the line connecting the centers of the touching circles.

Thus, the height of the third circle from line $l$ is $r_1 + r_2$.