Straight line intersecting $n$ parallel straight lines? 
Q) Through a fixed point $O$ a variable straight line $L$ is drawn to
  cut $n$ straight lines at points $P_{1},P_{2},P_{3}....P_{n}$  if
  $P$ is a point on the line $L$ such that OP is the H.M. of
  $OP_{1},OP_{2}.....OP_{n}$ . Find the locus of $P$ ?

I began by assigning parametric coordinates to find the length  $OP_{1},OP_{2}.....OP_{n}$ , but I'm getting equation in terms of $\frac{X}{Y}$ coordinates . How do I proceed ? Also shorter methods are always welcome .
 A: Let $y=mx$ be the equation of line $L$ (varying with $m$), while the equations of the $n$ lines $r_1, \ldots, r_n$ it cuts can be conveniently written as: $a_1 x+b_1 y =1$, $a_2 x+b_2 y =1$, ..., $a_n x+b_n y =1$. Finding the coordinates of $P_1$, …, $P_n$ is then straightforward:
$$
P_1=\left({1\over a_1+mb_1},{m\over a_1+mb_1}\right),
\ \ldots,
P_n=\left({1\over a_n+mb_n},{m\over a_n+mb_n}\right),
$$
so that
$$
OP_k={\sqrt{1+m^2}\over |a_k+mb_k|}.
$$
We then have
$$
{1\over OP}={1\over n}\sum_{k=1}^n{1\over OP_k}=
{1\over n\sqrt{1+m^2}}\sum_{k=1}^n{|a_k+mb_k|}.
$$
Suppose now that $m$ varies in an interval where the signs of all $a_k+mb_k$ don't change. In that case we can write $|a_k+mb_k|=a'_k+mb'_k$, where $a'_k=\pm a_k$ and $b'_k=\pm b_k$, the sign being chosen so that $a'_k+mb'_k\ge 0$. We have then 
$$
{1\over n}\sum_{k=1}^n{|a_k+mb_k|}=
{1\over n}\sum_{k=1}^n(a'_k+mb'_k)=\bar a + m\bar b,
$$
where
$$
\bar a={1\over n}\sum_{k=1}^n{a'},\quad \bar b={1\over n}\sum_{k=1}^n{b'}.
$$
It follows that
$$
OP={\sqrt{1+m^2}\over \bar a + m\bar b}
$$
so that $P$ is the intersection between $L$ and the line of equation
$\bar a x+ \bar b y =1$.
As long as all the quantities $a_k+mb_k$ don't change their sign while $m$ varies, point $P$ describes then a segment belonging to the line 
$\bar a x+ \bar b y =1$. But when $m$ comes to a value for which $L$ is parallel to some $r_i$, then $a_i+mb_i$ changes its sign, the values of $\bar a$ and $\bar b$ change, and thus $P$ describes a segment of a different line. 
The result is just a polygon with $2n$ sides: I leave to you understanding why this polygon has parallel opposite sides.
