If the points (p.q), (m,n) and (p-m, q-n) are collinear, show that pn=qm.

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Solution

Given points are collinear. Therefore,
$[p \times n + m (q - n) + (p - m) q] - [m \times q + (p - m) n + p (q - n)] = 0$
$(p n + q m - m n + p q - m q) - (m q + p n - m n + p q - p n) = 0$
$p n - m q = 0$
$p n = q m$

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If the points (p.q), (m,n) and (p-m, q-n) are collinear, show that pn=qm.

Given points are collinear. Therefore,[p×n+m(q−n)+(p−m)q]−[m×q+(p−m)n+p(q−n)]=0(pn+qm−mn+pq−mq)−(mq+pn−mn+pq−pn)=0pn−mq=0pn=qm

Given points are collinear. Therefore,
$[p \times n + m (q - n) + (p - m) q] - [m \times q + (p - m) n + p (q - n)] = 0$
$(p n + q m - m n + p q - m q) - (m q + p n - m n + p q - p n) = 0$
$p n - m q = 0$
$p n = q m$