000RM.dvi

118 Equilateral triangle in a rectangle

3.1 Equilateral triangle inscribed in a rectangle ....

Given a rectangleABCD, how can we choose a pointPonBCand a
pointQonCDsuch that triangleAP Qis equilateral?

P

Q

A B

D C

a

y

b−y

x a−x

b

SupposeAB=DC=a,BC=AD=b,DQ=x, andBP=y.
These satisfy

a^2 +y^2 =b^2 +x^2 =(a−x)^2 +(b−y)^2.

From these,2(ax+by)=a^2 +y^2 =b^2 +x^2 , and we have

(x^2 − 2 ax+b^2 )^2 =4b^2 (b^2 +x^2 )− 4 a^2 b^2.

This can be rewritten as

(x^2 +b^2 )((x^2 +b^2 )− 4 ax− 4 b^2 )=0,

from which
x=2a−

√

3 b.

Similarly,y=2b−

√

3 a.