1969 AMC 12 Problem 30

Question

Let

P

be a point of hypotenuse

AB

(or its extension) of isosceles right triangle

ABC.

Let

s=AP^2+PB^2.

Then:

s\lt2CP^2

for a finite number of positions of

P

s\lt2CP^2

for an infinite number of positions of

P

s=2CP^2

only if

P

is the midpoint of

AB

or an endpoint of

AB

s=2CP^2

always

s\gt2CP^2

if

P

is a trisection point of

AB

Put the midpoint of

AB

at the origin and

AB

on the

x

-axis

Use

A=(-a,0),

B=(a,0),

C=(0,a),

and

P=(p,0)

Take

A=(-a,0),

B=(a,0),

C=(0,a),

and

P=(p,0).

This describes every point on the hypotenuse line. Then

\begin{aligned} s&=(p+a)^2+(p-a)^2\\ &=2p^2+2a^2,\\ 2CP^2&=2(p^2+a^2)\\ &=2p^2+2a^2. \end{aligned}

Thus

s=2CP^2

for every position of

P.

Therefore, the correct answer is D.

Problem 30 in other years