#### Answer

The distance from the pitcher's position to first base is 63.7 feet.
The distance from the pitcher's position to second base is 66.8 feet.
The distance from the pitcher's position to third base is 63.7 feet.

#### Work Step by Step

Let A be the location of home plate.
Let B be the location of the pitcher's position.
Let C be the location of first base.
The points ABC form a triangle.
The angle $A = 45^{\circ}$, the side $b = 90~ft$, and the side $c = 60.5~ft$.
We can use the law of cosines to find $a$, which is the distance from the pitcher's position to first base:
$a^2 = b^2+c^2-2bc~cos~A$
$a = \sqrt{b^2+c^2-2bc~cos~A}$
$a = \sqrt{(90)^2+(60.5)^2-(2)(90)(60.5)~cos~45^{\circ}}$
$a = \sqrt{4059.857~ft^2}$
$a = 63.7~ft$
The distance from the pitcher's position to first base is 63.7 feet. By symmetry, the distance from the pitcher's position to third base is 63.7 feet.
The total distance from home plate to second base is $\sqrt{2}\times 90~ft$. The distance from the pitcher's position to second base is $\sqrt{2}\times 90~ft-60.5~ft = 66.8~ft$