Answer to Question #90476 in Trigonometry for Aravind

Question

If a triangle is equilateral, then the 3 medians are of equal length. Conversely if 3 medians of a triangle are of equal length, then the triangle is equilateral, prove.

Answer 1

Figure 1:

1.    AP=PC, AM=MB, BN=NC (Every triangle has exactly three medians, which divide any sides are equal in length).

2.    Now a centroid (point O) divides each median in the ratio 2:1. So we have 

AO:ON = CO:OM = BO:OP = 2:1

3.    The question says that the medians are equal in length. So, let us say:

If OP=ON=OM=a, then AO=CO=BO=2a;

Figure 2:

 

1.    The triangles AOP and BON are congruent by SAS (AO=BO, ON=OP, angleAOP=angleBON). 

Consequently, AP=BN.

2.   The triangles MOB and POC are congruent by SAS (AO=CO, OM=OP, angleCOP=angleBOM). 

Consequently, MB=PC.

 

3.   The triangles MAO and NAC are congruent by SAS (AO=CO, ON=OM, angleMOA=angleCON): 

Consequently, AM=NC.

 

4. AP=PC and AP=BN. Consequently, PC=BN.

BN=NC and BN=PC. Consequently, NC = PC. 

AP=PC=BN=NC. Consequently, AC=BC.

 

5.    AM=MB and AM=CN. Consequently, MB=NC.

BN=NC and NC=MB. Consequently, BN = MB. 

AM=MB=BN=NC. Consequently, AB=BC.

 

6.     AC=BC and AB=BC. Consequently, AB=BC=AC, so the triangle ABC is equilateral, that’s proved.