Given to prove that the medians of an equilateral triangle are equal
Median: The line joining the vertex and midpoint of opposite side.
Now, consider an equilateral triangle ABC
Let D,E,F are midpoints of , BC CA and AB
Then, AD, BE and CF are medians of ΔABC
Now,
Since ΔABC is an equilateral triangle
∠ABD = ∠BCE [From (3)]
[∠ABD and ∠ABC and ∠BCE and ∠BCA are same]
So, from SAS congruence criterion, we have
[Corresponding parts of congruent triangles are equal]
Now, consider ΔBCE and ΔCAF,
[ ∠BCE and ∠BCA and ∠CAF and ∠CAB are same]
CE = AF [From (2)]
So, from SAS congruence criterion, we have
[Corresponding parts of congruent triangles are equal]
From (4) and (5), we have
AD= BE= CF
Median AD = Median BE = Median CF
The medians of an equilateral triangle are equal
∴ Hence proved