New method of constructing external tangent to two circles?

[Gene Partlow]

I developed a way to construct an external tangent to two circles

that seems different from online examples such as at: http://jwilson.coe.uga.edu/EMAT6680Su06/Byrd/Assignment%20Six/RBAssignmentSix.html#:~:text=The%20Joining%20of%20Two%20Circles&text=When%20a%20line%20intersects%20a,through%20the%20point%20of%20tangency..

I haven’t found an example of my approach online (though I can’t

imagine that it is unknown.)  Explaining it w/o a diagram is fairly

cumbersome but I’ll try:

I draw two circles 1 and 2 of differing sizes [the external tangent to

circles of equal size is trivial to create].  With a straight edge, I draw

a line, c1-c2, joining their centers, c1 and c2, and extend it past the

smaller circle 1 some distance.  Then by construction, from circle

1’s center, c1, I draw a line segment perpendicular to the center-to

-center line c1-c2, such that the segment passes through circle 1

at point p1.  I do the same for circle 2, with both segments on the

same side of the center line.  So we have points of intersection p1

and p2, on circles 1 and 2 respectively.

I then draw a straight line containing p1 and p2 and extend it out to

meet the center-to-center line, at point p3.

By construction, I bisect the line segment p3-c1 at point p4. 

Using a compass, I draw an arc of the circle whose center is at

p4, with radius = p4-c1.  The intersection of this arc with circle 1 is

T1.  I connect p3 and T1 with a straight line segment.  This line

p3-T1 is tangent to circle 1, since the line segment T1-c1 is

perpendicular to the circle at T1, which is a definition of tangency

on the circle at T1.  

I extend tangent line p3-T1 to intersect circle 2, at T2.  This extended

line segment p3-T1-T2 is now tangent to both circles.  The logic of

this is seen more clearly if one views the two circles as parallel

slices through a right circular cone; or alternatively, as slices through

an infinite circular tube seen in perspective, p3 being its vanishing

point.

To restate, using the ‘tube-seen-in-perspective’ viewpoint:

A ray, Rc, extending from the vanishing pt. (p3), through the centers

c1 and c2, passes through all similar points c of any such circles .

And another ray Rp, also extending from p3 through p1 and p2,

passes through, and contains, all points p, as defined above.  In the

same way, another ray Rt, extending from p3 through T1, always

passes through, and contains, all points T similar to T1, ie: all points

defining tangency on all such circle ‘slices’, by a line containing p3.

Sorry for the verbosity…If I could show a diagram, it would be

quite obvious.

Gene