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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Partly Circles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Here are three problems involving circles.

Can you solve them and describe ways in which each might be connected to one or both of the others?







Firstly :
                     
Why does ab = cd?
                   
            (where a,b,c,d are lengths)







 
 
Secondly:

These three circles are drawn so that they touch each other and their centres are all on the line AB.
If CD is $8$ units in length, what is the area of the part that is shaded yellow?










Lastly:

If the area shaded yellow is equal to the area of the larger of the two circles that are shaded blue, what is the relationship between the radii of the three circles?