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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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Lens Angle

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Half a Triangle

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

This problem will need solving in stages by most students.

Perhaps beginning by establishing the required line ratio.

Then by remembering a familiar figure where that particular ratio can be found.

Deciding how that figure can be constructed on the given triangle to locate a useful point.

Finally creating the required line, parallel to the base, across the triangle.