Stage: 4 Challenge Level: 
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.
Mathematical reasoning & proof. Manipulating algebraic expressions/formulae. Other equations. Pythagorean triples. Circle theorems. Circles. Right angled triangles. Pythagoras' theorem. Creating expressions/formulae. Long problems.
Published October 1999.
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