The Enjoyment of Elementary Geometry Li Zhou ... - MAA Sections

È # #
1
#
!
(b) By the law of cosines, :œ + , #+, cos Ð Ñ and ;œ
È+ #
,#
1
#+, cos Ð ! Ñ can be constructed. Now, we notice that G œ È:; œ
#
:; :;
# # # #
ÉÐ Ñ# Ð Ñ# G E œ É+ #
,
#
G
#
F œ É+ #
,
#
G
#
. So we can construct , thus and as
well.
w
Finally, let # , in order to satisfy nTHE œ nTFG œ %& °. Therefore, JM Ÿ JEand
LN Ÿ LG, where M and N are the feet of the perpendiculars from J and L to GK and EI
respectively. If GK²IE, then JMŸJEœLN ŸLG œJM, which forces all the
equalities to hold. Thus MœHand NœF, and EFGHis a square. Otherwise, without loss of
generality, we may extend IE beyond E to intersect GK beyond K at O. Then JE LN Ÿ
LG, and therefore JM LN œ LG JE ÐLG JEÑsin nGOI LG JE. This
contradiction completes the proof.