We know Vermouth was disguised as Dr. Araide; this was because she recognized Sherry during the case in which Pisco appeared (Shiho was raised in the BO, so Vermouth had seen her even when she was a child), and realizing she might have turned into a child (she also saw Conan, and realized he was Shinichi since she was Yukiko's friend and saw photos of Shinichi's childhood) she decided to became the doctor in a school in order to approach to her and kill her.
Also, Jodie knew that Chris Vineyard was both her own mother (or more correctly, it was Sharon Vineyard that was also Chris Vineyard, who effectively never existed) and Vermouth; and since she went various times to Dr. Araide's surgery, she correctly thought her aim was murdering the doctor and take his place.
So... we know why Vermouth knew that Conan is Shinichi and that Ai is Sherry, and why Jodie knew that Dr. Araide was Vermouth. But... how did Conan realize Dr. Araide was Vermouth?
He only knew someone from the Black Organisation was near to Ai (Bus Hijacking, Ai has a "BO feeling"), and doesn't know about the existence of a "Vermouth" guy until he received the invitation for the Halloween Party. But we know Conan understood this, since it was him who told Jodie about the false report of the first Dr. Araide's case, in order to have Jodie prove that person wasn't the real doctor. So... how could he figure such a thing out?
