HYACINTHOS 21047

> From: Rabdy Hutson
>
> Dear friends:
>
> The incircle and Steiner inellipse intersect in 4 points, one of which, say P,
> is a triangle center, and the other 3, say A', B', C', form a
> central triangle (similar to the intersections of NPC and Steiner inellipse =
> X(115) + medial triangle).  P is a non-ETC center, search=2.307963780976356, and
> A'B'C' is perspective to ABC at non-ETC center, say X,
> search=1.626790309962831.
>
> Questions:
>
> 1.) What are the coordinates of P, A', B', C', and X?

Let t1=sqrt((c+a-b)(a+b-c)), t2=sqrt((a+b-c)(b+c-a)), t3=sqrt((b+c-a)(c+a-b)),
with t1,t2,t3 all > 0. Then P=(a-t1, b-t2, c-t3),
A'=(a-t1, b+t2, c+t3), B'=(a+t1, b-t2, c+t3),
C'=(a+t1, b+t2, c-t3) and X=(a+t1, b+t2, c+t3).

>
> 2.) For the ETC reference triangle, A' is the farthest of the 4
> intersections from A, and likewise for B' and C'.  Does this hold in
> general?
>
> 3.) For the ETC reference triangle, P is the closest of the 4 intersections to
> X(11).  Does this hold in general?
>
> 4.) Any other interesting properties?
>
> 5.) Generalizations for other inconics?
>
> Thanks in advance,
> Randy Hutson

--
Barry Wolk