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Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

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Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Shape and Territory

Age 16 to 18 Challenge Level:

If for a triangle $ABC$

$\tan(A - B) + \tan(B - C) + \tan(C - A) = 0$

what can you say about the triangle?