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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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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Far Horizon

Age 14 to 16 Challenge Level:

The observer's line of vision is along a tangent to the surface of the earth at the far point she can see on the horizon. Imagine a right angled triangle with the observer at one vertex, the far horizon point at another vertex and the centre of the earth at the third vertex.

You have enough information to find the angles in this triangle and hence the distance required.