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Prove that the medians of an equilateral triangle are equal.​

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The Triangle and Its Properties

Isosceles and Equilateral Triangles

Prove that the medians of a...

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Asked on December 20, 2019 by

Satyam Sangu

Prove that the medians of an equilateral triangle are equal.

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ANSWER

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 equ