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

Shape and Territory

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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?

Middle Man

Age 16 to 18
Challenge Level

Mark a point $P$ inside a closed curve.

Is it always possible to find two points that lie on the curve, such that $P$ is the mid point of the line joining these two points?