Step 1

Firstly, the metric should be defined. The properties must be remembered always while working on metric spaces.

The metric properties are, if d is a metric, given as:

$I)\text{}d\text{}\text{must be a real valued, finite and nonnegative.}$

$II)\text{}d\text{}(x,\text{}y)=0\text{}\text{if and only if}\text{}x=y$

$III)\text{}d\text{}(x,\text{}y)=d\text{}(y,\text{}x)$

$IV)\text{}d\text{}(x,\text{}y)\text{}\le \text{}d\text{}(x,\text{}z)\text{}+\text{}d\text{}(z,\text{}y)$

Step 2

Here, the metric should be defined as $d\text{}(x,\text{}y)$ must be the distance between the point x and y.

$d\text{}(P,\text{}Q)=\sqrt{{({x}_{1}\text{}-\text{}{x}_{2})}^{2}\text{}+\text{}{({y}_{1}\text{}-\text{}{y}_{2})}^{2}}$

$\text{Try to satisfy}\text{}d\text{}(P,\text{}Q)\text{}\le \text{}d\text{}(P,\text{}R)\text{}+\text{}d\text{}(R,\text{}Q)\text{}\text{where}$

$d\text{}(P,\text{}R)=\sqrt{{({x}_{1}\text{}-\text{}{x}_{3})}^{2}\text{}+\text{}{({y}_{1}\text{}-\text{}{y}_{3})}^{2}}\text{}\text{and}\text{}d\text{}(R,\text{}Q)=\sqrt{{({x}_{3}\text{}-\text{}{x}_{2})}^{2}\text{}+\text{}{({y}_{3}\text{}-\text{}{y}_{2})}^{2}}$

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