2016/quals/boomerang_constellations.html

<p>
The night sky can be modeled as an infinite 2D plane. There are <strong>N</strong> stars at distinct positions on this plane, the <strong>i</strong>th of which is at coordinates 
(<strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong>).
</p>

<p>
A boomerang constellation is a pair of distinct equal-length line segments which share a single endpoint, such that both endpoints of each segment coincide with a star's location.
</p>

<p>
Two boomerang constellations are distinct if they're not made up of the same unordered pair of line segments. How many distinct boomerang constellations can you spot?
</p>


<h3>Input</h3>
<p>
Input begins with an integer <strong>T</strong>, the number of nights on which you look out at the sky.
For each night, there is first a line containing the integer <strong>N</strong>. Then, <strong>N</strong>
lines follow, the <strong>i</strong>th of which contains the space-separated integers 
<strong>X<sub>i</sub></strong> and <strong>Y<sub>i</sub></strong>.
</p>


<h3>Output</h3>
<p>
For the <strong>i</strong>th night, print a line containing "Case #<strong>i</strong>: " followed by
the number of boomerang constellations in the night sky.
</p>


<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 50 <br />
1 &le; <strong>N</strong> &le; 2,000 <br />
-10,000 &le; <strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong> &le; 10,000 <br />
</p>



<h3>Explanation of Sample</h3>
<p>
On the first night, every pair of stars is a unique distance apart, so there are no boomerang constellations. On the second night, there are 4 boomerang constellations. One of them consists of the line segments (0,0)-(0,2) and (0,2)-(0,4).
</p>