Puzzle #223: A Walk in the Park

You decide to go for a walk in Puzzle Park. The park is perfectly square and its sides run north-south and east-west. A path runs from the southeast corner to the northwest corner, and another path runs from the southwest corner to the northeast corner. The paths meet at the center of the park.

You start walking from the southeast corner along the path to the center. You walk one-half of the distance to the center. The joy of stomping on grass appeals to you, and you walk due north for the same distance. You then turn left and walk the same distance again and are back on the southeast-northwest path. You walk the same distance yet again toward the northwest corner. The grass-stomping urge overcomes you again, and you walk due south the same distance again, turn left, and walk the same distance for the last time.

At this point, you are somewhere in the middle of the park. If the park is 350 meters on a side, how far are you from the exact center of the park?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, September 13, 2017 at noon Pacific Time. I will post the answer shortly after.