With Armstrong utterly confident of his own ability to hold the yellow jersey the New Yorker was given the green light to go for his own slice of the glory after infiltrating an early attack on the 205.5-kilometre (128-mile) stage. The 14-man break containing Hincapie and other riders not considered a threat to Armstrong overall slowly disintegrated as the race crossed no fewer than six Pyrenean climbs prior to tackling a 9.7km ascent to the Pla D'Adet ski station.
Half-way up the final climb, just Hincapie and the Spaniard Oscar Pereiro were left out at the head of the race, with the American craftily refusing to collaborate more than was absolutely necessary to maintain the duo's lead.
Pereiro made a feeble attempt to accelerate within sight of the finish, but Hincapie had calculated his strength perfectly to move ahead and become the eighth American in the history of the Tour to take a stage win.
"The original idea had been to get in the break so that Lance would have a rider waiting for him in support on the final climb." Hincapie said afterwards. "But when the move still had 10 minutes' advantage with 50km to go, it was clear that one of us guys ahead was going to win, and Lance told me to take my chance. It's like a dream come true."
Hincapie's look of utter amazement as he crossed the line was understandable: this is the first time since Armstrong began his run of six straight Tour wins that one of his team-mates has been able to take a stage.
In some ways Hincapie could not have been a more appropriate victor: he and Armstrong first met as teenagers and have raced on teams together since, and Hincapie is the only one of his team-mates to have been alongside Armstrong in all his six Tour wins.
Hincapie's win is not without its wider significance: for the Texan to allow one of his key mountain riders to take off in an early break on such a difficult stage, rather than keeping him in check in case of a crisis, reveals exactly how confident Armstrong is of his own strength.
Armstrong turned in another demonstration of raw power following attacks by the Italian contender Ivan Basso on the second last climb of the day, the Azet. Streaking across to Basso's back wheel, Armstrong then briefly moved ahead of the CSC rider, then to the side and ahead again - as a tactic of pure intimidation, it could hardly be bettered.
Flailing along behind came the German contender Jan Ullrich, who has finished second overall five times, but who was clearly suffering yesterday.
The T-Mobile leader clung on to Basso and Armstrong's shirt-tails for the remainder of the Azet, but threw in the towel on the final climb, when Basso once again put in a powerful acceleration and once again only Armstrong was able to follow.
The two had mutual interests at play in distancing their rivals: Basso to leapfrog over the Dane Mickael Rasmussen into second place overall, and Armstrong to widen his already considerable advantage over the remainder of the field.
The working alliance functioned to perfection, with the two gaining 1min 24sec over Ullrich and 1min 28sec over Rasmussen by the finish.
"It was the ideal end to a perfect day," Armstrong commented afterwards.
In fact, there is little left for the American to achieve as he enters his last week as a professional cyclist, although he has expressed a wish to win one final stage.
But as Hincapie found out yesterday, when circumstances permit the Texan is now in a mood to start distributing the largesse.
Alasdair Fotheringham writes for 'Cycling Weekly'Reuse content