GOING OFF ON A TANGENT

Venki Ramachandran

Here is an interesting challenge. How to construct a circle that is tangential to two given circles, one of which is inside the other.

To understand the scenario better, let us call them BIG CIRCLE, SMALL CIRCLE and TANGENTIAL CIRCLE. The SMALL CIRCLE is inside the BIG CIRCLE and we have t o construct the TANGENTIAL CIRCLE.

First of all, it is obvious that the TANGENTIAL CIRCLE is ALSO going to be inside t he BIG CIRCLE. If it is outside, then it won't be able to be tangential to the SMALLL CIRCLE without intersecting the BIG CIRCLE.

Next, we also need to recognize that as we can place the SMALLL CIRCLE anywhere inside the BIG CIRCLE, we probably will be able to construct more than one TANGENTIAL CIRCLE. In fact, we will see that we are going to be able to construct what is called, a FAMILY of such TANGENTIAL CIRCLES.

To construct such a TANGENTIAL CIRCLE it would help if we first identified the center of such a circle. The good news is that the centers of the BIG CIRCLE & the TANGENTIAL CIRCLE will be collinear. So we first draw a line from the center of the BIG CIRCLE and let it interect anywhere on the circumference of the BIG CIRCLE. For conveniece, let us ensure that such a line does not intersect the SMALL CIRCLE (although it does not matter as we will see in the GSP animation).

Now, the distance from the center of the TANGENTIAL CIRCLE to the outside of the SMALL CIRCLE (point of tangency) PLUS the radius of the SMALL CIRCLE, will have to be equal to the distance from the center of the TANGENTIAL CIRCLE to the point of tangency with the BIG CIRCLE PLUS the radius of the small circle. This is because the distance from the center of the TANGENTIAL CIRCLE to the two points of tangency (inside tangency of BIG CIRCLE and outside tangency of SMALL CIRCLE) will be the radius of the TANGENTIAL CIRCLE.

So we first identify a point that is on the line drawn from the center of the BIG CIRCLE to its circumference and extending beyond it such that its distance from the circumference of the BIG CIRCLE is equal to the radius of the SMALL CIRCLE. We then join this point to the center of the SMALLL CIRCLE.

Next, as we know that the center of the TANGENTIAL CIRCLE is going to be equidistant from this point and the center of the SMALL CIRCLE, we find the point of intersection of the perpendicular bisector of the the line joining this point and the center of the SMALL CIRCLE and the line that we drew from the center of the BIG CIRCLE to its circumference. This point of intersection is going to be the center of the TANGENTIAL CIRCLE.

Finally, CLICK HERE to see the locus of the center of the tangential circle.