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 CFG3) 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=DC2) 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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