Question

Prove that medians of an equilateral triangle are equal.

Answer 1

Let ABC be the equilateral triangle.
Then we have,
angle A=60°

angleB=60°

angleC=60°
and,
AB=BC=AC
and let AE , BD and CF be the medians.
A median divides a side into two equal parts.
AB=BC=AC

AF+BF=BE+CE=AD+CD

2AF=2BE=2AD

AF=BE=AD

therefore,
AF=BF=BE=CE=AD=CD............................1
In triangle AEC and triangle ABD we have.
AC=AB

angle C=angle A

EC=AD (from eq1)
By SAS congruency criterion we get,
triangle AEC congruent to triangle ABD.
By CPCT we get,
AE=BD..........................................................2

Similarly we can prove,
triangle ABD congruent to triangle AFC
Then,
BD=CF..........................................................3
By eq2 and eq3 we get,
AE=CF=BD
Hence proved that medians of an equilateral triangle are equal.

Answer 2

Part I. Part II.


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