Hi Alex,...

>AC + EF = CF + AE,

...

> What am I missing?

I think I've got it, after looking up various circumscribed quadrilateral properties to see which one you were referring to (my geometry is rusty in this area). Clearly you are referring to the theorem that the sums of opposite sides are equal for such quadrilaterals, but all the proofs I found showed the quadrilateral actually enclosing the circle. In the diagram you attached, the circle must clearly touch the 4 lines on the right side of the diagram. However, the lines do not enclose the circle. If lines AC and AE were tipped so that they met on the right at A', then the quadrilateral would actually surround the circle.

What you have here is a variation on the circumscribed quadrilateral, and I think you can account for the fact that it does not actually enclose the circle by thinking about directed line segments: the lines AC and AE "should" meet at the right in A', and the reversal causes the line segments AC and AE to reverse direction. Treating them as directed line segments gives

(-AC) + EF = CF + (-AE),

which correctly reduces to

AE + EF = AC + CF

as required by the theorem.

Mind you, I haven't actually checked that the quadrilateral theorem reverses signs like this, but I have a VERY strong hunch that this is the root of the trouble.

As to the general thrust of Dan Pedoe's comments, here is my way of thinking about why the circles are "inevitable" based on the pedagogy I use in teaching physics:

When an object is restrained at a point so that it cannot move, there must be a force, and if it cannot rotate, then there must be a torque to hold it'steady. Conversely, where a point is completely free to move, there is no force and where it is free to rotate, there is no torque. In short, where you do see motion, there is not constraint, and where you do not see motion, there is constraint. Obvious, but often overlooked by students.

In geometry similarly, when you see extension, there is no constraint. Circles extend around all angles from the center, so they do not constrain angle. All points are the same distance from the center, so circles show no extension in the radial direction, and so they do constrain distance. Lines, on the other hand extend to all distances from a given point, but with no variation in angle, hence they constrain angle but not length.

It is a kind of visual paradox: the aspect of the geometric object that you can see is its extension, but this constrains nothing, so has no logical content to be used in proof. The aspect you cannot see is what provides logical content. Hence with lines we see length but they are used in proofs related to angle, and with circles, we see angular extension, but they are used in proofs related to distance.

To this way of thinking, it is not surprising that circles are inevitable in a proof dealing with lengths of lines.

--Stuart Anderson