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Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.


Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?
The first two parts require only applying the idea that the tangents from an external point to a circle are equal and the radius is perpendicular to the tangent at the point of contact.

Possible approach
The first two parts could be tackled by the learners independently.

For the third part the formula for generating Pythagorean triples is needed.

Key questions
What facts do you know about tangents to circles?

What do you know about areas of triangles?

What is the formula for generating Pythagorean triples?

Possible support
Three articles: Pythagorean Triples I, Pythagorean Triples II and Picturing Pythagorean Triples.

Possible extension
The problem Pythagoras Mod 5
See the article Incircles which generalises this problem.