Apostolic constitution was a compilation of various sources. First generation followers of Christ pointed to Clement Romanus (70 AD) as primary source of the constitution,1 while liberal scholars mistakenly supposed its origin in 4 AD century.
In the book fifth section III on feast day and fast day, it is written:2
Brethren, observe the festival days; and first of all the birthday which you are to celebrate on the twenty-fifth of the ninth month.
The ninth month is relative to the starting date of Mary pregnancy, i.e. March 25.
March 25 + 9 months = December 25