Question 885787
Let {{{ a }}} = the girl's present age
Let {{{ b }}} = her brother's present age
---------------------------------
Starting at the end:
" when her age was half the sum of their present ages "
Her age was {{{ ( a + b )/2 }}}
---------------------------
" the age he was ( when her age was half the sum of their present ages ) "
Assume the brother is older
The difference in their ages will always be {{{ b - a }}}
So, " the age he was " is:
{{{ ( a + b ) / 2 + b - a = a/2 -a + b/2 + b }}}
{{{ ( a + b ) / 2 + b - a = (3/2)*b - a/2 }}}
-------------------------------------
Proceeding backwards:
" when she will be twice the age he was "
So she will be:
 {{{ 2*( (3/2)*b - a/2 ) = 3b - a }}}
-------------------------------
" the age her brother will be ( when she will be twice the age he was ) "
Add the difference in their ages to get the brother's age
{{{ 3b - a + b - a = 4b - 2a }}}
----------------------------
Proceed to the beginning:
" A girl's present age is ( the age her brother will be ) "
This tells me that
{{{ a = 4b - 2a }}}
{{{ 3a = 4b }}}
{{{ b = (3/4)*a }}}
So, the brother is not older
To be called a girl, she must be 
8, 12, or 16, which makes her brother
6, 9, or 12
( In order to have whole numbers )
That seems to be the best I can do