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 will make a second triangle with CF as a side from given triangle.
Step 1: Construct a line passing through F parallel to BE
Step 2: Construct a line passing through F parallel to AB
Step 3: Let's call the intesection as G. Then BEGF is to be a parallelogram and BE=FG.
Step 4: Construct a segment passing through C and G
What is E in the triangle CFG? ------ E is a centroid of CFG
1) AFH is similar to ABE and the ratio is 1:2
2) 2FH=BE and FG=BE(Step 3) therefore H is a midpoint of FG, i.e CH is a median of CFG
3) AH=HE and AE=EC therefore E is the point which is to be a 2:1 of from C to H, i.e E is a centroid
Investigate that ADCG is to be parallelogram and consider the reason
Why is ADCG a parallelogram?
1) FE=2EI and J is midpoint of EF and FE=DC therefore FE=JI and JI=DC
2) FE is parallel to DC therefore JDCI is parallelogram
3) AD=2JD and GC=2IC therefore AD=GC
Consider triangle ABE and triangle AFH.
They are similar with the ratio 2:1. So BE=2FH and AreaABE = 4 AreaAFH.
Area ABC = 2 Area ABE = 2Area AFC and so Area ABE = Area AFC
Area CFH = Area AFC - Area AFH = Area ABE - 1/4 Area ABE = 3/4 Area ABE
Since Area ABC = 2 Area ABE and Area CFG = 2 Area CFH,
Let's try to make sure by using measure in GSP for many cases.
Case 1. Equilateral triangl
Case 2. Isosceles triangle
Case 3. Right triangle
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