 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-iv/1/a/ii/solution

With

t

as in i, the foot of a perpendicular lies strictly between the endpoints precisely when

0 < t < 1

. Since the denominator is positive, this gives

0 < (c - a) \cdot (b - a) < ∥ b - a ∥^{2} .

Equivalently, both dot products

(c - a) \cdot (b - a) > 0, (c - b) \cdot (a - b) > 0

are positive: the angles of the triangle at

A

and

B

are both less than

π /2

. If “between” includes the endpoints, replace both strict inequalities by non-strict ones; equality gives

P = A

P = B

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