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Areas and Ratios

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

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?

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?