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Challenge 263: Point Positioning

Where should P be placed in this geometry problem?

ABC is a right-angled triangle with the right angle at A. P is a point on the hypotenuse BC, and points S and T lie on AC and AB respectively, such that PS is perpendicular to AC and PT is perpendicular to AB.

Where should P be placed on BC so that the length of ST is minimum?

Justify your answer clearly!