Let ABC be a triangle and A'B'C' the pedal triangle of I.
Denote:
(Oa), (Ob), (Oc) = the Soddy circles (A, AB'=AC'), (B, BC'=BA'), (C, CA'=CB'), resp.
1. The circle tangent to (Oa), (Ob), (Oc) internally touches them at A", B", C", resp.
(*) Carlos Hugo Olivera Díaz, PERU GEOMETRICO
Note: If ABC is not acute angled we take the largest circle tangent externally the circles (Oa), (Ob), (Oc)
(Oa), (Ob), (Oc) = the Soddy circles (A, AB'=AC'), (B, BC'=BA'), (C, CA'=CB'), resp.
As you say, these circles are called the A-, B-, C- Soddy circles, resp.
The circle tangent to (Oa), (Ob), (Oc) internally (externally)….
These circles are called the outer- (inner-) Soddy circles. Their centers are X(175) (X(176)) and their radius r^2*s/(r*(4*R+r)∓2*S), resp., where r, R, s, S are the inradius, circumradius, half-perimeter and double-area of ABC, resp.
The triangle with vertices the touchpoints of A- B- C- Soddy circles and the outer-Soddy circle is called the outer-Soddy triangle. Similarly, the triangle with vertices the touchpoints of A- B- C- Soddy circles and the inner-Soddy circle is called the inner-Soddy triangle.
The outer-Soddy triangle is perspective to the following triangles at the given perspectors: (ABC, 175), (3rd extouch, 10905), (4th extouch, 10908), (5th extouch, 10911), (Garcia-reflection, 15996), (intouch, 481), (7th mixtilinear, 10973), (inner-Soddy, 1)
The inner-Soddy triangle is perspective to the following triangles at the given perspectors: (ABC, 176), (3rd extouch, 10904), (4th extouch, 10907), (5th extouch, 10910), (Garcia-reflection, 15995), (intouch, 482), (7th mixtilinear, 10972), (outer-Soddy, 1)
For A”B”C” = the outer-Soddy triangle, I think a good name for the circle B’C’B”C” is the A-2nd-outer-Soddy circle of ABC
The triangle WaWbWc (centers of 2nd outer-Soddy circles) could be named the 2nd outer-Soddy triangle. Similar definitions for the A-2nd-inner-Soddy circle of ABC and the 2nd inner-Soddy triangle.
· Case I: WaWbWc = 2nd outer-Soddy triangle
Triangle WaWbWc and the following triangles are perspective with given perspector:
(ABC, 1), (Andromeda, 1), (anti-Aquila, 1), (Antlia, 1), (Aquila, 1), (BCI, 174), (2nd circumperp, 1), (4th Conway, 1), (5th Conway, 1), (excenters-midpoints, 1), (excenters-reflections, 1), (excentral, 1), (3rd extouch, 0), (4th extouch, 0), (5th extouch, 0), (incentral, 1), (intouch, 481), (inverse-in-incircle, 1), (Malfatti, 1), (medial, 0), (midarc, 1), (2nd midarc, 1), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear, 1), (7th mixtilinear, 0), (inner-Soddy, 0), (outer-Soddy, 481), (inner-Yff, 1), (outer-Yff, 1), (inner-Yff tangents, 1), (outer-Yff tangents, 1),
where perspector 0 means a not-ETC center.
Triangle WaWbWc and the following triangles are orthologic with given orthologic centers:
(4th anti-tri-squares, 0, 0), (Ascella, 1, 0), (Atik, 1, 0), (1st circumperp, 1, 0), (2nd circumperp, 1, 0), (inner-Conway, 1, 0), (Conway, 1, 0), (2nd Conway, 1, 0), (3rd Conway, 1, 0), (3rd Euler, 1, 0), (4th Euler, 1, 0), (excenters-reflections, 1, 0), (excentral, 1, 6212), (2nd extouch, 1, 0), (hexyl, 1, 0), (Honsberger, 1, 0), (inner-Hutson, 1, 0), (Hutson intouch, 1, 0), (outer-Hutson, 1, 0), (incircle-circles, 1, 0), (intouch, 1, 481), (inverse-in-incircle, 1, 0), (6th mixtilinear, 1, 0), (2nd Pamfilos-Zhou, 1, 0), (1st Sharygin, 1, 0), (outer-Soddy, 175, 481), (tangential-midarc, 1, 0), (2nd tangential-midarc, 1, 0), (3rd tri-squares, 0, 0), (Ursa-major, 1, 0), (Ursa-minor, 1, 0), (outer-Vecten, 0, 6212), (Wasat, 1, 0), (Yff central, 1, 0), (2nd Zaniah, 1, 0)
where 0 means a not-ETC center.
Triangle WaWbWc and 2nd Sharygin are parallelogic.
· Case II: WaWbWc = 2nd inner-Soddy triangle
Triangle WaWbWc and the following triangles are perspective with given perspector:
(ABC, 1), (Andromeda, 1), (anti-Aquila, 1), (Antlia, 1), (Aquila, 1), (2nd circumperp, 1), (4th Conway, 1), (5th Conway, 1), (excenters-midpoints, 1), (excenters-reflections, 1), (excentral, 1), (3rd extouch, 0), (4th extouch, 0), (5th extouch, 0), (incentral, 1), (intouch, 482), (inverse-in-incircle, 1), (Malfatti, 1), (medial, 0), (midarc, 1), (2nd midarc, 1), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear, 1), (7th mixtilinear, 0), (1st Pamfilos-Zhou, 0), (inner-Soddy, 482), (outer-Soddy, 0), (inner-Yff, 1), (outer-Yff, 1), (inner-Yff tangents, 1), (outer-Yff tangents, 1)
Triangle WaWbWc and the following triangles are orthologic with given orthologic centers:
(3rd anti-tri-squares, 0, 0), (Ascella, 1, 0), (Atik, 1, 0), (1st circumperp, 1, 0), (2nd circumperp, 1, 8225), (inner-Conway, 1, 0), (Conway, 1, 0), (2nd Conway, 1, 0), (3rd Conway, 1, 0), (3rd Euler, 1, 0), (4th Euler, 1, 0), (excenters-reflections, 1, 0), (excentral, 1, 6213), (2nd extouch, 1, 0), (hexyl, 1, 0), (Honsberger, 1, 0), (inner-Hutson, 1, 0), (Hutson intouch, 1, 0), (outer-Hutson, 1, 0), (incircle-circles, 1, 0), (intouch, 1, 482), (inverse-in-incircle, 1, 0), (6th mixtilinear, 1, 0), (2nd Pamfilos-Zhou, 1, 8225), (1st Sharygin, 1, 0), (inner-Soddy, 176, 482), (tangential-midarc, 1, 0), (2nd tangential-midarc, 1, 0), (4th tri-squares, 0, 0), (Ursa-major, 1, 0), (Ursa-minor, 1, 0), (inner-Vecten, 0, 6213), (Wasat, 1, 0), (Yff central, 1, 0), (2nd Zaniah, 1, 0)
Triangle WaWbWc and 2nd Sharygin are parallelogic.
I will calculate and check the non-ETC centers as soon as I have time.
César Lozada
[Peter Moses]:
Hi Antreas,
Another set of circles passes through the B&C vertices of the outer and inner Soddy triangles & cyclic.
centers, T = {-(1/sa),1/sb,1/sc} & cyclic.
squared radii = (5 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2)/(4 (3 a^2-2 a b-b^2-2 a c+2 b c-c^2)^2) & cyclic.
The triangle T, simple and somewhat surprisingly unlisted, is perspective to lots of triangles ...
For example, the perspector of the incentral triangle and T is
= X(1)X(9446)∩X(2)X(3119)
= (a + b - c)*(a - b + c)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^3*c + a^2*b*c - a*b^2*c - b^3*c - 2*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : := lies on these lines: {1, 9446}, {2, 3119}, {7, 354}, {55, 658}, {57, 14189}, {77, 614}, {85, 3742}, {192, 6168}, {241, 2275}, {279, 9445}, {347, 23668}, {348, 24477}, {390, 9533}, {664, 1376}, {883, 27538}, {1442, 20277}, {1462, 2162}, {1996, 3475}, {3158, 25716}, {3599, 8236}, {3673, 17626}, {3676, 23655}, {4124, 17090}, {4388, 7055}, {4566, 5281}, {5226, 27475}, {5274, 10004}, {5437, 9312}, {7176, 24268}, {7182, 10453}, {17084, 18633}, {21609, 30947}
= X(1)-Ceva conjugate of X(7)
= X(1742)-cross conjugate of X(3177)
= crosspoint of X(1) and X(1742)
= barycentric product X(i) X(j) for these {i,j}: {7, 3177}, {57, 20935}, {85, 1742}, {331, 20793}, {664, 21195}, {1434, 21084}, {6063, 20995}}.
= barycentric quotient X(i) / X(j) for these {i,j}: {{1742, 9}, {3177, 8}, {20793, 219}, {20935, 312}, {20995, 55}, {21084, 2321}, {21195, 522}, {21856, 210}
Perspector of the extouch triangle and T ...
= X(2)X(3160)∩X(7)X(354)
= (a + b - c)*(a - b + c)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 4*a^3*c - 4*a^2*b*c + 4*a*b^2*c + 4*b^3*c + 6*a^2*c^2 + 4*a*b*c^2 - 10*b^2*c^2 - 4*a*c^3 + 4*b*c^3 + c^4) ::= anticomplement of the isogonal of X(1419)
= X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 10405}, {56, 20059}, {57, 9812}, {109, 3239}, {144, 3436}, {165, 329}, {1419, 8}, {3160, 69}, {3207, 144}, {7339, 658}, {9533, 3434}, {16284, 21286}, {17106, 7}
= X(8)-Ceva conjugate of X(7)
= X(i)-cross conjugate of X(j) for these (i,j): {2124, 17113}, {2951, 30695}
= X(i)-isoconjugate of X(j) for these (i,j): {6, 2125}, {55, 8917}
= cevapoint of X(i) and X(j) for these (i,j): {2124, 2951}, {3160, 15913}
= barycentric product X(i) X(j) for these {i,j}: {7, 30695}, {8, 17113}, {75, 2124}, {85, 2951}, {1615, 6063}, {4554, 17427}
= barycentric quotient X(i) / X(j) for these {i,j}: {1, 2125}, {57, 8917}, {1615, 55}, {2124, 1}, {2951, 9}, {17113, 7}, {17427, 650}, {30695, 8}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {479, 497, 7}, {1088, 10580, 7}, {7056, 9812, 7}
Peter Moses.
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