Three circles of radius 6, 7, and 8 are externally tangent to each other. There exists a smaller circle tangent to all three (in the space created between the three original circles).

What is the radius of this smallest circle?

Well, frankly, the Pythagoras theorem will do it. Give a second, in the process of calculating...

Drawing to scale will not be helpful as the radius of the smallest circle is unknown, as of course, the one we are asked to find!

So, wait, after moving on, it is not the Pythagoras theorem that will solve it, instead, the Cosine-Rule, with the condition that the angle of the point (the centre of the smallest circle) is 360º.

So, wait, I am back with the calculation...

UGH! It is futile with the cosine there!

Anyway, I will be back!