Summing products in spaced columns

I'm sure the following is possible, but cant quite figure it out... Say theres 3 column headings, repeated across columns multiple times. There are multiple rows of data, and I am trying to create a summary statistic for each row. The longhand calculation I need to do is A3*B3/$A$1 + D3*E3/$A$1 + G3*H3/$A$1 + ... for however many columns I have. The first column heading data (columns A, D, G...) is just a number, as is the second heading (columns B, E, H...). $A$1 is the sum of all the second heading values (B3+E3+H3+...).
One way to get the value for $A$1 is {=SUM((MOD(COLUMN(B3:L3)-COLUMN(B3),3)=0)*(B3:L3))}, where column L is the end of the range of interest (and is actually more like column ED, hence I dont want to do the longhand).
I sense this should be possible with SUMPRODUCT, but suspect one of my problems is that the above sum only works as an array formula, which I cant make work with the SumProduct.

Anyways, can anyone tell me how to achieve this with or without the SUM and SumProduct functions, such that I can easily generate the summary statistic? As well as there being many additions to make longhand, I also need to make several different summary statistics so it would take ages.

Example data attached.

Thanks, jsw

Hello,
The calc is averaging a bunch of averages, but because the data is calculated from discrete values, a weighting is required for each of the data. You could think of each set of three columns as being data from a month, for example, where the average 10 (e.g. col A) is derived from e.g. 3 data points (col B) in January. If I included 12 months of data, simply averaging the 12 averages in this situation does not give the correct answer (i.e. does not give the same answer as if I took all the raw data and averaged it); weighting by the number of data points in each month does. Hope that makes sense?

Beautiful, Thank you Mr Wilson. Very much appreciated. Once question about your solution - the second array is offset by one column (B:M) - I understand that the array needs to be the same size as the others, A:L, however what prevents the slightly modified solution below from working (second array in SumProduct)? Must the first element in the returned array be a 'hit', which is achieved by using B:M but not A:L?
=SUMPRODUCT(--(MOD(COLUMN(A3:L3),3)=1),A3:L3,--(MOD(COLUMN(A3:L3),3)=2),A3:L3)/SUMIF(A2:L2,"counts",A3:L3)
Thanks again for your help, jsw