As shown in the diagram, two semi-circles (each of radius $1/2$) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles, and find a 3-4-5 triangle in the diagram.

Regular polygons and circles. Polynomial functions and their roots. Circle properties and circle theorems. Creating and manipulating expressions and formulae. Sine rule & cosine rule. Pythagoras' theorem. Mathematical reasoning & proof. 2D shapes and their properties. Simultaneous equations. Other equations.