Complete the following proof.

Prove: In an equilateral triangle the three medians are equal.

Step-by-step explanation:

In the figure attached,

ΔABC is an equilateral triangle,

Sides AB = BC = AC and points P, Q, and R are the midpoints of these sides respectively.

If the coordinates of A(0, 0), B(2a, 0) and C(a, b)

AB = 2a

AC =

Since AB = AC

2a =

4a² = a² + b²

3a² = b²

Therefore, ordinate pairs representing midpoints of AB, BC and AC will be

P = =(a, 0)

Q = =

R = =

Now we will find the lengths of medians with the help of formula of distance between two points (x, y) and (x', y')

d =

AQ =

     =

     =

     =  [Since b² = 3a²]

     =

BR =

     =

     =

     =

     =

CP = b =

Therefore, AQ = BR = CP =

Hence, medians of an equilateral triangle are equal.

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Good luck!

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