Dog & Gopher

A large field has a dog and a gopher. The dog wants to eat the gopher, while the gopher wants to run to safety through one of several gopher holes dug in the surface of the field.

Neither the dog nor the gopher is a math major; however, neither is entirely stupid. The gopher decides on a particular gopher hole and heads for that hole in a straight line at a fixed speed. The dog, which is very good at reading body language, anticipates which hole the gopher has chosen, and heads at double the speed of the gopher to the hole, where it intends to gobble up the gopher. If the dog reaches the hole first, the gopher gets gobbled; otherwise, the gopher escapes.

You have been retained by the gopher to select a hole through which it can escape, if such a hole exists.

The first line of input contains four real numbers: the $(x,y)$ coordinates of the gopher followed by the $(x,y)$ coordinates of the dog. Subsequent lines of input each contain two real numbers: the $(x,y)$ coordinates of a gopher hole. All coordinates are in metres, with exactly three digits after the decimal point.

There are not more than $1000$ gopher holes and all coordinates are between $-10\, 000$ and $+10\, 000$.

Your output should consist of a single line. If the gopher
can escape the line should read “`The
gopher can escape through the hole at ($x$,$y$).`” identifying the
appropriate hole.

Otherwise the output line should read “`The gopher cannot escape.`” If the gopher may
escape through more than one hole, any one will do. If the
gopher and dog reach the hole at the same time, the gopher
escapes successfully.

Sample Input 1 | Sample Output 1 |
---|---|

1.000 1.000 2.000 2.000 1.500 1.500 |
The gopher cannot escape. |

Sample Input 2 | Sample Output 2 |
---|---|

2.000 2.000 1.000 1.000 1.500 1.500 2.500 2.500 |
The gopher can escape through the hole at (2.500,2.500). |