at his side for an Aug. 13 homecoming party, the cyclist, 32, spoke with PEOPLE.
Engagement rumors were flying about you and Sheryl after the Tour. So: Gotten engaged lately?
[laughs] No! I don't know how these things get started. Sheryl and I sometimes sit back and laugh at the stuff we read about us.
How are things going?
Very well. We're happy. We just spent two weeks in Florida with the kids [Luke, 4, and twins Isabelle and Grace, 2; Armstrong and wife Kristin were divorced in 2003]. They're really fond of her. She's very talented and amazing. Things are going really great.
As a father, how tough is it to dedicate yourself to months of training?
Incredibly difficult. I was gone for two months in the spring, came home for six weeks and was gone again for three months. Being separated, you miss too many things. They're changing all the time, and I don't want to miss it. I won't train that way again. From now on I'll alter my schedule so I'm not away so long.
What did they think of your win?
[laughing] The girls were very angry when I took off my helmet. They said, "Daddy's naughty. He took off his helmet." They did not like that.
Several friends said they connected your win to your battle against cancer. Was it that way for you?
For me, all the wins are connected to my beating cancer. Cancer survivors get to share in these wins, and it gives them hope. That will be my biggest regret when I stop racing: that it won't be a July festival of cycling and comeback for people who struggle, like me, against cancer.
The Lance Armstrong Foundation is branching out into lobbying efforts and supporting enrollment in clinical trials. What's next?
It's not my goal for us to be the biggest nonprofit in the land, but I want us to make the most difference. We aren't just cancer survivors. To live and thrive and have an extraordinary life is another thing. It's just like the yellow wristband says: "LIVESTRONG."
Lance Armstrong will always have Paris after all, he won his record-breaking sixth Tour de France there last month—but there's no place like home. Returning to Austin with his three kids and girlfriend