Where do you find the 3rd after you find her in front of the contest stadium?

That gym leader you saw, Fantina, is actually the fifth gym leader. You have a while to go before you can battle her.

Go to Veilstone City and beat the gym leader there. Once you beat her, exit and talk to Dawn/Lukas. They tell you to meet them outside of Team Galactic's Warehouse in the northwest corner of town. So, heal up, and go meet them. Help them defeat Team Galactic to get their Pokedex back, and once you've done that, remember to go in the Warehouse to get HM02 Fly! Now, go to Pastoria City and battle the gym leader THERE. Once you beat him, heal up, and go in the Safari Zone (northeast corner of town). Once you're in, talk to a guy to your right and obtain HM05 Defog (this is useful for an event that is coming up). Now, exit, and talk to the Galactic grunt facing the building. Start chasing him, and beat your rival, and then continue chasing him. Heal up, if necessary. Now, once you've chased him into the Valor Lakefront, be prepared to battle him. Once you beat him, go up a little and talk to Cynthia. She'll give you the Secretpotion. With this, as she told you, you have to go to Route 210 (north of Solaceont Town) and give that to the Psyduck. Go there and do that, and Cynthia will come up again and give you the Old Charm. She tells you to go to Celestic to give that to her grandmother. Well, now that the path is cleared, you can now go to Celestic Town! How convenient! At first, you'll notice the path is clear. But if you keep going up, you'll be walking in dense fog. This is where HM05 Defog comes in handy. Use that to clear the path, and follow the route until you reach Celestic Town. Heal up if necessary, and go in the middle of town, talking to an old lady along the way, and beat the Galactic grunt. The old lady will come up again, notice the Old Charm and tell you that she's Cynthia's grandmother, and takes the Old Charm from you. She tells you to go in the cave, so do just that, and go all the way in the back and examine the drawing. She'll come up once again, and give you HM03 Surf, and tells you that in order to use it, you need to obtain the Hearthome Gym Badge. Once again, convenience has occured, because now you can obtain the fifth badge in Hearthome City!

The third gym is in veilstone. The girl with the purple hair is the 5th gym leader.

I'm guessing you are talking about Fantina? First of all, go to Pastoria City. In front of the Great Marsh there should be one of the team galactic grunts. Follow him from pastoria to the lake and then he will battle you and leave. Then you go to the lake closest to pastoria, name escapes me for the moment, and then go battle the leader, either jupiter or saturn I think. (you have to talk to the grunt for him to leave.)