33456번 - Old Solution Methods 스페셜 저지다국어

There are $6$ distinct points $A$, $B$, $C$, $D$, $E$, $F$ on the plane. Points $A$, $B$, and $C$ are not collinear.

Svetozar drew a line $a$ through points $A$ and $D$, a line $b$ through points $B$ and $E$, and a line $c$ through points $C$ and $F$. It turned out that none of these lines are parallel. Now he wants to rotate line $a$ around point $A$, line $b$ around point $B$, and line $c$ around point $C$ counterclockwise by the same angle, then find the intersection points of the resulting lines, and then draw a triangle with vertices at the obtained three points (if they are not collinear).

Svetozar wants to obtain a triangle with the largest possible area. If it is not possible to form a triangle, Svetozar considers the area to be zero. Find this area.

The first line contains a single integer $T$ ($1 \leq T \leq 10^5$), denoting the number of test cases.

Then $T$ descriptions of test cases follow. Each description consists of $6$ lines, describing points $A$, $B$, $C$, $D$, $E$, $F$ respectively. Each point description consists of two integers $x$ and $y$ ($-20 \leq x, y \leq 20$): the coordinates of the point.

It is guaranteed that in each test case all points are distinct, points $A$, $B$, and $C$ are not collinear, and the lines $AD$, $BE$, and $CF$ are pairwise not parallel.

For each test case, output a single real number with an absolute or relative error not exceeding $10^{-6}$: the largest possible area of the triangle. It is guaranteed that in none of the tests this area exceeds $10^7$.

In the example, the maximum possible area is achieved by rotating the lines by an angle approximately equal to $104.036^\circ$. The triangle with the largest area has sides equal to $\sqrt{17}$, $\sqrt{17}$, $\sqrt{34}$, and vertices at points $(a, b)$, $(a - 1, b + 4)$, and $(a + 4, b + 1)$, where $a \approx 3.8235$, $b \approx 1.7059$.

In the illustration below, the original lines are marked with dashed lines, and the lines after rotation are marked with solid lines: