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.

Pythagoras' theorem. Mathematical reasoning & proof. Other equations. Plane shapes and their properties. Cartesian equations of circles. Creating expressions/formulae. Similarity and congruence. Regular polygons and circles. Circle properties and circle theorems. Manipulating algebraic expressions/formulae.