The common tangents in D, E, F are the radical axes of these circles pairwise,
They then concur in a single point I, radical center of these circles.
ID = IE = IF and these lines being perpendicular to the sides of ABC, I is the incenter and DEF the contact points with incircle
Hence the existence and uniqueness of the three circles, mutually externaly tangent
(See appendix for internally tangent circles)
There are two such circles. (P') corresponds to the searched outer Soddy circle (P).
The other one (P") is the image of the inner Soddy circle.
The contact point U of (P) and (A) is the image of U' and is therefore on EU'.
And the same for the contact points V and W with (B) and (C).
Finally P, E and P' are in line, as the circles (P) and (P') are inverse of each other.
Hence a construction of the outer Soddy point and circle :
Construct the contact points DEF of the incircle, and circle (A) with center A going through D (and F)
The perpendicular from A to BC intersects this circle in U', opposite to BC, and in U" near BC.
Let P' the symmetric of A with respect to U'
Let U the other intersection of EU' with circle (A)
Lines AU and EP' intersect in P, center of the outer Soddy circle.
U is the contact point of (A) with this circle, hence the outer Soddy circle is the circle centered in P, going through U.
We do the same for the inner Soddy circle, starting from U", P".
As this construction becomes here very inaccurate, we may use the following variant :
Consider also the point V", image of the contact point with (B) in an inversion through pole F :
The perpendicular from B to AC intersects circle (B) in V", opposite to AC.
Let Q" the symmetric of B with respect to V"
Let P" the symmetric of A with respect to U" as above,
The center S of the inner Soddy circle is the intersection of EP" and FQ".
The contact point U" is sent back to U, other intersection of EU" with (A) (not drawn on the figure)
And finally the inner Soddy circle is the circle with center S, through U
Let the circle (P), centered in P, tangent to the three circles (A) (B) (C), outer Soddy circle, which, if the circles
(A) (B) (C) are not too much unequal, surounds these circles.
The circles (A) (B) (C) being internally tangent in U, V and W.
We get : PU = PV = PW = RP radius of the Soddy circle
CW = CE and BV = BE
The perimeter of PCB is then PV + PW = 2RP
And the same for the two other triangles
Theorem : The outer Soddy point is the isoperimetric point
This only if the outer Soddy circle (P) surrounds the circles (A) (B) (C).
If the smaller of circles (A) (B) (C) is so small that (A) (B) (C) touch (P) externally, there is no isoperimetric point.
The critical value is when the outer Soddy circle is a straight line :
UV² = AB² - (BU-AV)² = (rA + rB)² - (rA - rB)², hence UV = 2√(rArB)
and the same for VW, then UW = UV + VW gives a condition for existence of the isoperimetric point :
1/√rA < 1/√rB + 1/√rC or also :
|a + b + c > 4R + r
with r the inradius and R the circumradius.
The inner Soddy point is similarily a point of "equal detour".
Define the detour of A to B through S as the value AS + SB - AB.
If S is the inner Soddy point, the detour is 2ρ with ρ being the radius of the inner Soddy circle, and this is the same with AB, BC and AC.
Finally, if the outer Soddy point is not an isoperimetric point, it is another point of equal détour (when a + b + c < 4R + r).
The radii of the Soddy circles are given by the
Descartes formula, from the radii of circles (A), (B), (C) that is
(well known formulas about the contact points of incircle)
circle (A) : rA = AD = AE = (AB + AC - BC)/2 = s - a with s being the semi-perimeter of ABC
and the same for the two others.
|The Soddy line goes through the incenter|