Euclid: The Game - Level 23: Construct an external common tangent to two given circles in five moves? In Euclid: The Game - Level 23, we are given two non-concentric circles: 



The goal is to create one of the tangents shown in the following image:



Note: it suffices to create a line segment on one of these $2$ tangents. This segment need not actually touch both circles, it just needs to be in the right direction. 

Part of the challenge in Euclid: The Game is to do this with a minimal number of "moves", each move standing for the usage of one of the tools given. Creating a point or naming an intersection do not count as a move; everything else does. 

The "minimum number of moves" to solve this is said to be $6$. Multiple solutions exist (see below). However, the page mentions that "there are rumours that it is possible to do this level in 5 non-primitive moves".

So the question, is: can someone come up with a solution in $5$ (non-primitive) moves?



SPOILERS BELOW!

I know of two different solutions in $6$ moves:

Solution 1:


    
 Define a random point $M$ on the circle $c_B$. 
 Translate the line $BM$ to $A$, creating a segment $AN$. (move 1) 
 Define the intersection $O = c_A \cap AN$.
 Compass the length $ON$ centered at $B$, this creates a circle $c_B'$. (move 2)
 Create the midpoint $P$ of $AB$. (move 3)
 Create the circle $c_P$ centered at $P$ with radius $PB$. (move 4)
 Define  $Q = c_P \cap c_B'$.
 Create the ray $BQ$. (move 5)
 Translate $AQ$ to $R$, creating a segment $RS$. (move 6)


Solution 2:


   
 Define a random point $M$ on the circle $c_B$. 
 Translate the line $BM$ to $A$, creating a segment $AN$. (move 1) 
 Define the intersection $O = c_A \cap AN$.
 Create the ray $BA$. (move 2)
 Create the ray $MO$. (move 3)
 Define $P = BA \cap MO$.
 Create the midpoint $Q$ of $AP$. (move 4)
 Create the circle $c_Q$ centered at $Q$ with radius $PQ$. (move 5)
 Define  $R = c_Q \cap c_A$.
 Create the ray $PR$. (move 6)




Remarks:


Both $6$-moves-solutions start similarly by picking a random point $M$. Perhaps picking a not-so-random point (for example on the intersection of the circles, see next item) to start with can reduce the number of points needed?
You can move the centers of the circles, for example creating intersecting circles! In some of the levels, this is necessary to get the minimum number of moves, so perhaps it can be of use here too?
Here is a simple one in 5 moves -


Draw two parallel lines through centers of the circles - 2 moves
Draw a line through two intersections (on same side) - 1 move
Draw the perpendiculars to the chords through the centers, in the two circles - 2 moves


The intersections of these two perpendicular lines with the perspective circles are defining the segment as required in the question - those are actually the tangent points on the two circles.

I do not think a drawing is needed.
Here is a simple one in 5 moves -


Draw two parallel lines through centers of the circles - 2 moves
Draw a line through two intersections (on same side) - 1 move
Draw the perpendiculars to the chords through the centers, in the two circles - 2 moves


The intersections of these two perpendicular lines with the perspective circles are defining the segment as required in the question - those are actually the tangent points on the two circles.

I do not think a drawing is needed.