HYACINTHOS 25582

[Antreas P. Hatzipolakis]:

Excentral Triangle version of a problem (orthic triangle version) by Hình Học [ = Dao Thanh Oai]  (*)

Let ABC be a triangle and A'B'C' the antipedal triangle of I (excentral triangle).

Denote:

(Ja), (Jb), (Jc) = the incircles of A'BC, B'CA, C'AB, resp.

The circle tangent externally the circles (Ja), (Jb), (Jc) touches internally (I)

Which is its center and which is its point of contact with (I) ?

Variation:

Denote:

(Jaa), (Jbb), (Jcc) = the excircles of A'BC, B'CA, C'AB corresponding to A', B', C', resp.

I think the circle tangent internally (Jaa), (Jbb), (Jcc) touches internally (I).

Which is its center and which is its point of contact with (I) ?

 

(*) Posted to FB group BÀI TOÁN HAY - LỜI GIẢI ĐẸP - ĐAM MÊ TOÁN HỌC

[Peter Moses]:

Hi Antreas,

 

>(Jaa), (Jbb), (Jcc) = the excircles of A'BC, B'CA, C'AB corresponding to A', B', C', resp.

>I think the circle tangent internally

(Jaa), (Jbb), (Jcc) touches internally (I).

>Which is its center and which

is its point of contact with (I) ?

The center is:

a (a^4 b+2 a^3 b^2-2 a b^4-b^5+a^4 c-6 a^3 b c+4 a^2 b^2 c-2 a b^3 c+3 b^4 c+2 a^3 c^2+4 a^2 b c^2-2 b^3 c^2-2 a b c^3-2 b^2 c^3-2 a c^4+3 b c^4-c^5-4 b c ((2 a^3+a^2 b-2 a b^2-b^3+a^2 c-4 a b c+b^2 c-2 a c^2+b c^2-c^3) Sin[A/2]-a ((a^2-b^2+6 a c+c^2) Sin[B/2]+(a^2+6 a b+b^2-c^2) Sin[C/2])))::

on lines {3,6585} and {1, touchpoint}.

searches {1. 62544145241393729697238901601, 1. 72846232377694886518246049526, 1. 69383297202467421323554982916} .

 

Best regards,

Peter Moses.