If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$,then prove that $AC = \frac{1}{2} AB$. Explain by drawing the figure.

Which of the following statements are true and which are false? Give reasons for your answers.
$(i)$ Only one line can pass through a single point.
$(ii)$ There are an infinite number of lines which pass through two distinct points.
$(iii)$ $A$ terminated line can be produced indefinitely on both the sides.
$(iv)$ If two circles are equal,then their radii are equal.
$(v)$ In the figure,if $AB = PQ$ and $PQ = XY,$ then $AB = XY$.

Give a definition for parallel lines. Are there other terms that need to be defined first? What are they,and how might you define them?

Give a definition for the radius of a circle. Are there other terms that need to be defined first? What are they,and how might you define them?

Consider two 'postulates' given below:
$(i)$ Given any two distinct points $A$ and $B$,there exists a third point $C$ which is in between $A$ and $B$.
$(ii)$ There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid's postulates? Explain.

Does Euclid's fifth postulate imply the existence of parallel lines? Explain.