Suppose a circle (O,R) and a point A within its area having distance r from O (0<=r<=R). Which points X of the circle minimize the angle between the line AX and the tangent on X?
I was pondering on "realistic" (take this with many grains of salt) ways an energy shield could stop incoming projectiles without stopping outcoming ones. So I was led to think of a one-way high air drag setup. The drag would be "outwards"; thus, incoming bullets would hit it dead on and suffer an 100% penalty; but if they hit the shield at an angle, they would only be dragged back by the cos of that angle. The question then becomes, if you are an outward attacker, what is the best angle to attack someone inside.
In the following image, the question is: which are the points X that minimize the angle φ.