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

[UltraMarginal]

I got part way through the article, and will read the rest later.

I think the trick to the question is 'real life'

If you are travelling at a constant speed and you are not effected by the need to maintain a coefficient of friction with the road surface, the fastest way around a corner is the shortest distance. no questions asked. imagine that track had two cars on it that were travelling at a steady 10km an hour, neither car will skid in the turn but if one is on the outside of the track and one is on the inside of the track, the car on the inside has less distance to travel and will finish the turn faster. it's the same reason why they have staggered starts in the running races in the Olympics.

if however you are driving a car around a corner as fast as you can and have maximum speeds you can travel because you must maintain solid contact with the road at all times, the best path is essentially as straight a line as you can manage through the corner.
The more you try to turn the car the more force is applied to the tyres in a sideways direction, eventually if this force gets too high the tyres will skid. This is related to the coefficient of friction of the tyres against the specific road surface, speed, direction and weight of the car.

The simplest description is, you have to slow down for a corner right? So you make that corner as straight as you can.

[MV75]

Friction does not hold your car to the surface of the road. The ability of the tyre to key into the surface is grip.

The shortest way isn't the fastest way. Doing what is essentially a U turn by hugging the inside is the slowest way possible. The fastest way through a corner is the straightest way with minimal turning. The straightest way is to go wide, then hit the apex and exit wide. This is something you've already even demonstrated to him. I'd just not bother if he is still going to argue it.

[Trent]

Remember the racetrack "Silence" on F-Zero-X?

That track answered this question pretty well