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In the adjoining figure, we have X and Y are the mid-points of AC and BC and AX = CY. Show that AC = BC.

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Given, X is the mid-point of AC

AX = CX= ½ AC

⇒ 2AX =2CX = AC …(i)

Y is the mid-point of BC.

BY = CY = ½ BC

⇒ 2BY = 2CY= BC …(ii)

According to the question,

We also have,

AX=CY …(iii)

Applying the Euclid’s axiom,

“Things which are double of the same things are equal to one another”.

We get,

From Eq. (iii),

2AX = 2CY

Using Eqs. (i) and (ii), we get,

AC=BC

Hence Proved.

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