I have a follow up question for a previous question linked here:What is the equation for the distance between these two circles?
In that question asked for an equation to find the distance between the a random point in the green circle’s perimeter and the point on the blue circle that the perpendicular line of the tangent line of the green circle’s point intersects(The circles I am talking about are in the last two images.). I found a pretty good answer but later found a better solution.
Well after finding the solution I started trying to find the correlation between the point in the green circle’s y-coordinate and the distance. And I found that they are the exact same. To show that, that means the purple line and the black line in the image at the bottom of this question are equal.
My question is why. Why are they equal? Is there something that already shows this correlation if so what is it?
I know a lot of people are probably going ask for the equations I used and stuff, and if you want it is in the other question. I don’t think the reason is going to be found in deriving the equation though.
For example the Pythagorean theorem of a squared plus b squared equals c squared. Lots of people know the formula and can see it’s true, but a lot of people don’t know why it’s true. But after some close analysis of the square below, you can see why it is true.
Let the equation of the green circle be $r=R$, and the equation of the red circle be $r=R \,\cos \theta$.
Let the point $A$ on the big circle be $(R\,\cos\theta,R\,\sin\theta)$.
Then the magnitude of the segment $AB$ will be $R-R\,\cos \theta$, and the perpendicular dropped from $A$ to the line $x=R$ will have magnitude $R-R\,\cos\,\theta$.