000RM.dvi

330 Constructions with the golden section

Exercise

1.LetABCbe an equilateral triangle. The line joining the midpoints

D,Eof two sides intersects the circumcircle atF. ThenEdivides

DFin the golden section,i.e.,

DE

DF

=

√

5 − 1

2

.

D

B C

A

F

E

2.M is the midpoint of the sideABof a squareABCD. The line

DMintersects the circle with diameterABat two points,Pinside

andQ outside the square. Show that the rectangleAP BQis a

golden rectangle,i.e.,PB:PA=(

√

5+1):2.

A B

D C

M

P

Q