Construct a square PQRS such that lines PQ, QR, RS, and SP pass through given points A, B, C, and D respectively.

Discuss the configuration of the given points when the construction is not possible.

Given points A(x1,y1) B(x2,y2) C(x3,y3) and D(x4,y4)
we have to construct a square => find the slope(m) of 
line passing through A is enough to construct a square

y-y1=m(x-x1):L1
y-y2=-1/m(x-x2):L2
y-y3=m(x-x3):L3
y-y4=-/m(x-x4):L4
Distance between parallel lines(L1,L3) and (L2,L4) 
should be same as it is a square.

=>|(y1-y3)-m(x1-x3)/√(1+mē)|=|(y2-y4)+1/m(x2-x4)/√|+1/mē|
=>|(y1-y3)-m(x1-x3)|=|m(y2-y4)+(x2-x4)|

Case(i): m=(y1-y3-x2+x4)/(y2-y4+x1-x3)
Case(ii): m=(y1-y3+x2-x4)/(x1-x3+y4-y2)

the construction is not possible if neumarator and denominator
are both zeroes (indeterminate form)
=> y1+x4=x2+y3 -- eq(1) &
=> y2+x1=y4+x3 -- eq(2)

Posted by Praneeth on 2008-05-30 04:14:35