Advanced High-School Mathematics

42 CHAPTER 1 Advanced Euclidean Geometry

3. LetA, B, X, andY be colinear points. Define thecross ratioby
   setting

[A,B;X,Y] =

AX

AY

·

Y B

XB

(signed magnitudes).

Show that the colinear points A, B, X, and Y are in harmonic

ratio if [A,B;X,Y] =−1.

4. Show that for colinear pointsA, B, X, andY one has

[A,B;X,Y] = [X,Y;A,B] = [B,A;Y,X] = [Y,X;B,A].

Conclude from this that under the 4! = 24 permutations ofA, B, X,

andY, there are at most 6 different values of the cross ratio.

5. LetA, B, X, andY be colinear points, and setλ= [A,B;X,Y].
   Show that under the 4! permutations ofA, B, X, andY, the pos-
   sible values of the cross ratio are

λ,

1

λ

, 1 −λ,

1

1 −λ

,

λ

λ− 1

,

λ− 1

λ

.

6. If A, B, X, and Y are in a harmonic ratio, how many possible
   values are there of the cross ratio [A,B;X,Y] under permutations?


7. LetAandBbe given points.

(a) Show that the locus of points{M|MP = 3MQ}is a circle.

(b) LetXandY be the points of intersection of (AB) with the cir-

cle described in part (a) above. Show that the pointsA, B, X,

andY are in a harmonic ratio.

8. Show that if [A,B;X,Y] = 1, then eitherA=BorX=Y.


9. Theharmonic meanof two real numbers is aandbis given by
   2 ab
   a+b
   . Assume that the pointsA, B, X,andY are in a harmonic