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.