How do you explain to laymen the fastest way around a corner using maths?

[Borgeman]

Here is a solid explanation, and compares the apex line with the inside of the track and the outside.
http://phors.locost7.info/phors05.htm
Its an idealised scenario but should satisfy what you're looking for. It's all about aiming for the highest average speed through a corner.

But if you're trying to convince someone of this who has watched what pro racing drivers do and still believes otherwise, put him in the bin cause he is a lost cause.