trigonometry - Raycasting "fisheye effect" question

Continuing my exploration of raycasting, I am very confused about how the correction of the fisheye effect works.

Looking at the screenshot below from the tutorial at permadi.com, the way I understand the cause of the fisheye effect is that the rays that are cast are distances from the player, rather than the distances perpendicular to the screen (or camera plane) which is what really needs to be displayed. The distance perpendicular to the screen then, in my world, should simply be the distance of Y coordinates (Py - Dy) assuming that the player is facing straight upwards.

Continuing the tutorial, this is exactly how it seems to be according to the below screenshot. From my point of view, the "distorted distance" below is the same as the distance PD calculated above, and what's labelled the "correct distance" below should be the same as Py - Dy. Yet, this clearly isn't the case according to the tutorial. My question is, WHY is this not the same? How could it not be? What am I understanding and visualizing wrong here?

UPDATE: Here's another perspective. The tutorial at lodev.org has another way of handling the fisheye effect which confuses me in the same way. This one relies on distance vectors more than angles and calculates the perpendicular distance to the wall according to the below formulas where mapX is the position of the player, rayPosX is the position of the wall that has been hit by a ray, and rayDirX is the direction of the ray (along with its Y counterparts of course). (1-stepX)/2 is simply a way of adding 1 if the ray is on the left side of the field of view. By the same logic as in the above tutorial, the perpendicular distance is simply mapX - rayPosX in my world. Why does rayDirX need to be divided to it?

//Calculate distance projected on camera direction (oblique distance will give fisheye effect!)
if (side == 0)
      perpWallDist = fabs((mapX - rayPosX + (1 - stepX) / 2) / rayDirX);
else
      perpWallDist = fabs((mapY - rayPosY + (1 - stepY) / 2) / rayDirY);

Answer

I understand it well enough now. What I described in the question only applies when facing perfectly upwards/downwards or to the left/right. When the player faces any other direction than this however, you of course have to convert the X or Y distance to the larger, real perpendicular distance of the camera plane, hence the additions to the formulas. I knew it was simple!