Independent arc length sum

Some things just never change … like the total length of the arcs AB and CD in the figure below, if the chords are able to slide back and forth:

independent arcs 1

That is, as long as the angles between the lines don’t change, arc length AB + arc length CD stays constant, irrespective of the position of X within the circle.

Here is my proof:

Consider what happens when you slide X along AC to a new point X’; let the line through X’, parallel to line BXD, meet the circle at points B’ and D’, with B’ on the same side of AC as B, and D’ on the same side of AC as D, as shown:

independent arcs 2

Now lines BDD’B’ is an isosceles trapezium as it is cyclic; so arc BB’ has the same length as arc DD’.

So how has moving X to X’ changed the sum of arc length AB + arc length CD?

Arc length AB has decreased by arc length BB’; arc length CD has increased by arc length DD’; so arc length AB + arc length CD has not changed.

Now we could’ve slid X along BD as well, and arc length AB + arc length CD  would not have changed for the same reasons; therefore, by using a combination of these two moves, we can move X anywhere in the circle, and arc length AB + arc length CD will still be the same.

I give credit for this result-ette to Geoff Smith, although this is my own proof. It can also be done by considering the proportions of the angles at the centre.

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