Triangle reflections

Blue Triangle ABC has midpoints D, E, F.
The medians were drawn and used as reflection lines.
Drag A, B, and C. Ttry to make the reflected triangles concurrent with the original triangle.

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Can you make two of the triangles concurrent?
Three of the triangles concurrent?
All of the triangles concurrent?
Explain.

Linda Johnson, August 5, 2008, Created with GeoGebra