Subject: 1/3rd of the way from 35N,106W to 13N,100E (great circle)

I'm flying on the great circle from 35N,106W to 13N,100E.

Where am I one third of the way along my journey?

More generally, where am I at r% of my journey, and for arbitrary
(nonantipodal) start and end points.

Some sites claim you can find the "through the Earth" line joining
these two points, go one third of the way along this line, and then
project back to the sphere, but that's inaccurate.

Question

Subject: 1/3rd of the way from 35N,106W to 13N,100E (great circle) 
 
I'm flying on the great circle from 35N,106W to 13N,100E. 
 
Where am I one third of the way along my journey? 
 
More generally, where am I at r% of my journey, and for arbitrary 
(nonantipodal) start and end points. 
 
Some sites claim you can find the "through the Earth" line joining 
these two points, go one third of the way along this line, and then 
project back to the sphere, but that's inaccurate.