Task A
Take any triangle ABC. Construct equilateral triangles externally on each
side and locate the centroid of each equilateral triangle. Label these centroids
A', B' and C' for the triangle centroids opposite angles A, B and C.
Construct lines AA', BB' and CC'. Observe. Are they concurrent? Is the point
of concurrency any of the orthocenter, centroid, incenter or circumcenter
of triangle ABC? If not what can you find out about this point.
By the nature of the construction of this sketch we already know a few
things. Triangle A'B'C' is the Napoleon
triangle and is thus an equilateral triangle. The point P (the cuncurrency
of lines AA', BB' and CC') is discussed fully in Task
D though the fact that the lines are concurrent is not proved!
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