 In this video, we are going to prove that when two tangent circles are present like this, their point of contact and their centers are collinear. So let's say this is first circle and this is the second circle and these two circles are tangent to each other. That means there exists a tangent like this which is tangent to both the circles at the same time. Let us name these two points and a contact point. So let's say this is point A and this is point B and name this point here as M. Let us also name the tangent as P, Q and what we need to prove is that A, M and B are collinear. Now let us try to analyze this situation. Let us say we join A, M and B, M and I am not going to draw it here but I will first draw it here for you like this. So let's say this is A and this is M and this is P. We can see in this diagram that A, M, B are not collinear. They do not lie on a single line. What are the properties of a line like this? You take any angle on the line it should be 180 degrees but here if they are not collinear, this angle here will be less than 180 degrees or you can have a figure like this where A, M and B, M join like this and the angle here would be greater than 180 degrees. But we need to select this case here and if any angle that we select is 180 degrees, then we will be sure that the given points will be on the single line. So let us first join A, M and B, M. So now I have joined A, M and B, M. They do look like they are collinear but we still want to prove that. We know that PQ is the tangent to the circle with center A and is also a tangent to the circle with center B. And so therefore, A, M and B, M are radii of first and second circles respectively. So now we will just quickly write that PQ is tangent to both the circles and therefore from the property that the radius is always perpendicular to the tangent, we can say that A, M is perpendicular to PQ and therefore angle A, M, P. This angle here, angle A, M, P is equal to 90 degrees. Also B, M is perpendicular to PQ and therefore angle B, M, P is also 90 degrees. Now what is the sum of these two angles? Angle A, M, P plus angle B, M, P. Now angle A, M, P plus angle B, M, P gives us the angle A, M, B, right? Angle like this. If they were not collinear, this would be the angle that we will be talking about. So this is A, then M and this is B, right? But since both are 90 degrees, this is 90 degrees, this is 90 degrees. This must be 180 degrees and therefore this is 180 degrees and if the angle is 180 degrees, we must have a straight line and therefore all the points A, M and B lie on a single line and therefore we have proved that points A, M and B are collinear. Now we could have the circles case like this as well where one of the smaller circle is inside the other circle and still PQ is a tangent to both the circles. Now in this case, we can see that A, M is perpendicular to PQ and B, M is also perpendicular to PQ from the property of circles since B is the center of the smaller circle and A is the center of the larger circle. What we need to prove here is that points A, B and M are collinear. From these two facts, we know that angle A, M, P is equal to 90 degrees as well as angle B, M, P is equal to 90 degrees. Now consider this case. If points A, B and M were not collinear, we would have a diagram like this. So this was PQ something like this and we would have drawn A and M and if point B was not on line A, M, we would have had something like this. So this was B here. If there was some angle X here. Now if angle A, M, P is 90 degrees, I can write that from the given figure. If B was not on line A, M, angle B, M, P should be 90 plus X degrees. 90 degrees plus X degrees. But I also know that angle B, M, P is 90 degrees and that makes X degrees to be equal to 0 and since X degrees is equal to 0, there is no deviation of segment B, M from segment A, M and the angle between A, M and B, M is 0 degrees and therefore B lies on the line A, M and therefore A, B and M are collinear and this is how we can prove that.