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#81 Right triangles built in equilateral triangle

This was confusing to me--see remarks below


The segments are concurrent only for a special case, it seems to me. Are the big triangles congruent? Where exactly is the point in the equilateral triangle? The points E, G, H cannot be independently moved since clearly if we move, say, G leaving E, H fixed the segments would no longer be congruent. If anyone has the book and has some insight it would be helpful to hear. Thanks.