this little part right here is nothing other than z minus x and now we have our first connection by papa pythagoras namely that r squared plus and then we are going to get z minus x squared is equal to z plus x squared and do you know what's even cooler than that papa pythagoras actually gives us the chance to get the difference of two squares what we're going to do is we are going to subtract this part on both sides giving us that r squared is nothing other than z plus x squared minus z minus x squared and now this right here is the difference of two squares obviously factoring this is going to give us okay first part plus the second one so z plus x plus z minus x this is good something's going to cancel out and next up we are going to take this part minus what we got in here so times set plus x minus set and then plus x our negative is going to turn into a positive and this is good that's nice simplification x and x is going to cancel out here giving us NUMBER_TOKEN z and here that is going to cancel out giving us NUMBER_TOKENx meaning overall r squared is nothing other than NUMBER_TOKEN times two is going to give us four four times x times z and now taking the square root on both sides the positive branch at that because r is a positive side length in the euclidean space gives us overall that r is nothing other than square root of four is two we can use the multiplicative property of the square root to break it up basically because x and said that product is also positive NUMBER_TOKEN times the square root of x times that that's a pretty cool relationship between r and these radii now we are going to go through the same construction with the other circles pairwise let us go ahead and take the tiny nut and a tiny little bit bigger nut next now what we are going to have is we are going to have one right here okay the circle with radius NUMBER_TOKEN and then that's a good circle that's not a good circuit that's an ellipse or something of that sort now what we're going to do is once again we are going to connect our centers to one another by the same logic as before since they are connected and they are passing through the arcs of their tangents basically okay we are going to get that this length right here is going to be x once again plus one so this right here is x plus one now next up we are going to form another right triangle we are going to do the same thing as before giving us right triangle in here now we are going to call this length after r comes s and now what we are going to do is we are going to c yet again since this line goes downwards just like before okay this is this line we are going to get that this is just the radius x minus the small side length y that we are having here meaning this right here is x minus one and once again we are going to get the same situation as we did up here the smaller side length is of the form x minus another something or we had z minus x right here meaning overall if we were to go through the same process once again for s we are going to get that s is nothing other than NUMBER_TOKEN times
NO_SEGMENT_TOKEN