 Hello and welcome to the session. In this session we will discuss about touching a tangent circle and common tangents. If the two circles touch each other at a single point, the circles tangent circles to the same line at the same point. You can see these two circles which centers O and O dash touch each other at a single point P. So they are the tangent circles or you can say touching circles. Also we observe from the figure that these two circles are tangent to the same line S, T and at the same point P. So they are the tangent or touching circles. Then to each of the two circles which centers O and O dash exterior or you can say the two common tangents to the two circles, the circles lie on the same side of the common tangent and so the tangents are the exterior or direct. We have the two circles which do not intersect each other. We have four tangents to both the circles. They lie on the same side of these tangents. Consider these two tangents. Then you can see that the two circles lie on the opposite side. The two circles are on the same tangent. We equal to R1 and the radius of the smaller circle. Then this will be equal to radius of the bigger circles R1 minus the radius of the smaller circle which between the centers and on the opposite sides of there that there are three lines into these two circles. This line is also a tangent to these two circles and even this line is also a tangent to these two circles. V R1 that is we have O dash P is equal to R1. Radius of the circle with center R, V R2 is D between the centers of the two circles that is R dash is equal to R1 plus each other externally. Then the distance between the centers is radii. We have already did each other. In this case these two lines would be called exterior common tangents. When there would be four these two would be exterior common tangents and these two would be interior common tangents in which one circle is contained in another circle with center all the two circles touch each other externally which is shown by figure one and when the two circles touch each other internally shown in figure two. To the theorem we have to the two circles lies on the straight line joining the centers of the two circles. Tangent Pt to the two circles of the two circles with the point of contact of the two circles to each other. The radius of the two circles that is Pt is the perpendicular to Pt and O dash P is the perpendicular to Pt in both the cases. Is equal to 90 degrees is also equal to 90 degrees. So we have got these angles of measure 90 degrees. Figure one when the two circles touch each other externally plus angle O dash Pt this would be equal to 90 degrees plus 90 degrees which is equal to 180 degrees. The two circles touch each other externally. We consider the case when the two circles touch each other internally is equal to Pt as each is equal to 90 degrees and that O dash P lies along O P means that P lies on the straight line through O and O dash touch each other internally or it is supposed to prove on tangents.