Note:
- In acute triangle, orthocentre (H), centroid (G), and circumcentre (O) are collinear and
- Centroid of the triangle formed by points
,
,
is
.
- If the circumcentre of a triangle formed by
,
and
is origin, then its orthocentre is
(using 1).
Example 1:
Find the relation if ,
,
,
are the points of the vertices of a parallelogram taken in order.
Solution:
As the diagonals of a parallelogram bisect each other, the affix of the mid-point of AC is same as the affix of the mid-point of BD. That is,
or
Example 2:
if ,
,
are three non-zero complex numbers such that
where
then prove that the points corresponding to
,
, and
are collinear.
Solution:
latex \frac{(1-\lambda)z_{1}+\lambda z_{2}}{1-\lambda +\lambda}$.
Hence, divides the line joining
and
in the ratio
. Thus, the given points are collinear.
Homework:
- Let
,
,
be three complex numbers and a, b, c be real numbers not all zero, such that
and
. Show that
,
,
are collinear.
- In triangle PQR,
,
, and
are inscribed in the circle
. If
be the orthocentre of triangle PQR, then find
.
More later,
Nalin Pithwa