$A$ line is a length without $\ldots \ldots \ldots$

Given two distinct points,there is a ......... line that passes through them.

Solve the following question using appropriate Euclid's axiom:
In the figure:
$BM = BN$,$M$ is the mid-point of $AB$ and $N$ is the mid-point of $BC$. Show that $AB = BC$.

Solve the following question using an appropriate Euclid's axiom:
In the figure,$X$ and $Y$ are the mid-points of $AC$ and $BC$ respectively,and $AX = CY$. Show that $AC = BC$.

Write whether the following statement is True or False. Justify your answer.
The things which are double of the same thing are equal to one another.

In ancient India,altars (or vedis) with combinations of shapes like rectangles,triangles,and trapeziums were required for $\ldots \ldots \ldots$