A little proof of the area relationship between a given triangle and its triangle of medians

(I did this proof on Oct. 16, 2009, and translate it into technology language (my 6th finished write-up) on Oct. 19 Monday)

Chen Tian

Original problem:

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.

Solution:

If we think about one of the popular properties of the "subtriangles" of any triangle, we may remember that some of them have the same area, based on the condition of having equal bases and same height. So I decide to try to show the area relationship between a given triangle and its triangle of medians.

You can use the GSP tool of measuring the area of any triangle to make a conjecture about the area relationship, and you will find that the area of the triangle of medians of a given triangle is 3/4 of the area of the given triangle. It would be a good start of exploration for students with the help of technology.

By the way, here is the script tool of constructing the triangle of medians of a given triangle.

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