## maximising the angle to view a portrait at the gallery.

How is the angle $\theta$ affected by the distance, x, away from the portrait?

But is this model realistic? What does maximising angle $\theta$ really mean? Does making the angle largest give the clearest view of the picture? would the person standing distance x away from the portrait not tilt his head?

A nice geometry argument claims that the maximum angle occurs when a circle draw that passes the bottom and top of the portrait is tangential to the eye level.

This is because if the circle passing through the 3 points (the eye, bottom and top of the portrait) intersects the eye level at 2 distinct points, the angle $\theta$ can be further maximise when choosing to stand at a point between the 2 distinct points cut by the circle.

It turns out that the above modelling is similar to modelling for the best position of a soccer player to kick the ball from the side line.