HYACINTHOS 26874

[Kadir Altintas]:

Let ABC be a triangle and (Oi), i=1,2,3,4 four congruent circles such that:

(O1) touches AB, AC and (O4)
(O2) touches BC, BA and (O4)
(O3) touches CA, CB and (O4)

Let P, Q, R be the touchpoints of the circles (O1), (O2), (O3) and (O4), resp.

The incenter of PQR lies on the OI line.

[Peter Moses]:

 

 

Hi Antreas,

P = {(a-b-c) (2 a^2-b^2+2 b c-c^2),b^2 (-a+b-c),-(a+b-c) c^2}.
O4 = X(35)
O1 = {a^4-2 a^2 b^2+b^4-a^2 b c-2 a^2 c^2-2 b^2 c^2+c^4,-a b^2 c,-a b c^2}.
I' = midpoint X(1) X(35) = X(2646).
radius of the 4 circles = r R / (2 r + R).
radius incircle of PQR = r^2 / (2 r + R).

Best regards,
Peter Moses.

 

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[Kadir Altintas]

Thank you, Mr. Peter Moses

ABC, PQR are perspective.

Perspector?

 

 

 

 

[Peter Moses]:

 

 

Hi Antreas,

>ABC, PQR are perspective.

X(55).

Perspective to lots of triangles.
For example:
medial at X(5432).
anticomplementary at X(5218)
Euler at X(5217).

Paralogic with the Brocard triangle at:
(a-b-c) (2 a^6-3 a^4 b^2+2 a^2 b^4+2 a^4 b c-2 a^2 b^3 c-3 a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 b c^3+2 b^3 c^3+2 a^2 c^4-b^2 c^4)::
on lines {{2,12354},{3,3027},{11,620},{20,12184},{35,2782},{55,99},{98,5217},{114,6284},{115,5432},...}

and X(6).

Many orthologics too.

Best regards,
Peter Moses.