Let's replace the first, second, third, and home with letters.
first base = F
second base = S
Third base = T
home base = H
We have two triangles, ΔHTS and ΔHFS
We know HS = HS because it's a shared side.
We know FS = TS
We know that ∠HST and ∠HSF are equal.
Because of SAS, the two triangles are congruent.
According to CPCTC, TH must be congruent to FH.
Thus, the distance from home plate to third base and the distance from home plate to first base must be the same.
Have an awesome day! :)