From The Student Room

TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP I 1997 question 5 solution

---

(Here, for not-getting-killed-by-LaTeX reasons, I'll use to denote vectors . Furthermore, I won't write out the dot product operator symbol, so you should just assume means . And, I'll use the notation to mean .)

Let a, b, c, d be the position vectors of points A, B, C, D, respectively. Then position vectors of P, Q, R, S are (a + b)/2, (b + c)/2, (c + d)/2, (d + a)/2, respectively.

To prove:

We shall prove the equivalent form

.

The LHS can be written in vector form as

The RHS is

and so LHS = RHS, and we have proven

.

Now, since only depends on the lengths of the rods, and the rods are rigid, the expression is going to stay constans as the rods move.

Similarly, we shall prove

Written in vectors, this is

LHS:

RHS:

and so LHS = RHS, and

Since now only depends on the lengths of the rods, it will remain constant. QED.

(Actually, there are two angles formed by AC and BD and depending on which one you choose, you'll end up with different values of (one of the values is minus the other). But as long as you stick with the same angle, it's going to be the same scalar product, and so the value remains constant.)

Solution by ukgea