Question based on chords of a circle

Question

Question:

Given a circle and two points $P$ and $Q$ not neccessarily on that circle. Perpendiculars are drawn from points $P$ and $Q$ to the polar lines of the points $Q$ and $P$ respectively. Prove that the ratio of of lengths of those perpendiculars are equal to the ratio of distances of point $P$ and $Q$ from the centre of circle.

Attempt:

I solved this question by assuming the circle equation as $x^2+y^2=1$

Let $P$ be $(x_1,y_1)$ and $Q$ be $(x_2,y_2)$.

Polar of $P$ is $$xx_1+yy_1-1=0$$ Polar of $Q$ is $$xx_2+yy_2-1=0$$

The perpendicular distance from $P$ to polar of $Q$ is $$\frac{x_1x_2+y_1y_2-1}{\sqrt{{x_2}^2+{y_2}^2}}$$ Similarly, perpendicular distance from $Q$ is $$\frac{x_1x_2+y_1y_2-1}{\sqrt{{x_1}^2+{y_1}^2}}$$ Ratio is $$\frac{\sqrt{{x_1}^2+{y_1}^2}}{\sqrt{{x_2}^2+{y_2}^2}}$$which is the ratio of distance from centre of circle $(0,0)$ to the points $P$ and $Q$.

Is there any method to solve this using geometry? It looks like a problem involving two similar triangles.

Answer 1

I was staring at this$ \space\downarrow$

for nearly a week, and couldn't come up with anything 'elegant' because I was searching for similar triangles. Then I drank a few cups of coffee and remembered something:

$\square AGIO\sim \square BHJO$ . Quadrilaterals can be similar too.