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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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Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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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?

Impossible Triangles?

Age 16 to 18 Challenge Level:

You might first like to try the slightly easier problem Equal Equilateral Triangles to get a feel for the possibilities.

Will the areas of the shapes be rational or irrational? What lengths of sides would be able to give you these areas?