Construct a
triangle and its medians. Construct a second triangle with the three sides having
the lengths of the three medians from your first triangle. Find some
relationship between the two triangles. (E.g., are they congruent? similar?
have same area? same perimeter? ratio of areas? ratio or perimeters?) Prove
whatever you find.
I started constructing a triangle ABC and its medians AD, BE, and
CF. 

Then I traced a line at point C, parallel to the median BE, and
another line at E, parallel to the side BC. These two lines have one point of intersection G. The segment CG has the same length
and orientation than the median BE. 


Then I traced a line at C, parallel to the median AD, and another
line at A, parallel to the side BC.
These two lines have one point of intersection H. The segment CH has the same length
and orientation than the median AD. 



I continued tracing a line at point G, parallel to the median CF,
which contains point H. The
segment GH has the same length and orientation than the median CF. 



The sides of the triangle CGH have respectively the same lengths
than the medians AD, BE, and CF of the triangle ABC. 

This animation shows how unique is triangle
CGH. If triangle ABC changes,
triangle CGH changes also. 




This animation shows that there is a constant
relationship between the areas of both triangles: the area of the triangle of
medians is threefourths of the area of the given triangle. 




This animation shows that the ratio of the
perimeter of the triangle to the perimeter of the given triangle never is
less than .75 or more than 1.0 







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