HYACINTHOS 21048

Dear friends,

[Randy Hutson]

> > 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',
> > Questions:
> > 1.) What are the coordinates of P, A', B', C', and X?

[Barry Wolk]

> 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).

[Randy Hutson]

> Generalizations for other inconics?

If the isotomic conjugates of the persectors of two
inconics are interior points of ABC with
barycentric coordinates (pp : qq : rr), (PP : QQ : RR)
then their intersections are
S = ( (Qr-qR)^2 : (Rp-rP)^2 : (Pq-pQ)^2 )
A'= ( (Qr-qR)^2 : (Rp+rP)^2 : (Pq+pQ)^2 )
B'= ( (Qr+qR)^2 : (Rp-rP)^2 : (Pq+pQ)^2 )
C'= ( (Qr+qR)^2 : (Rp+rP)^2 : (Pq-pQ)^2 )
and the triangles ABC, A'B'C' are perspective at
X = ( (Qr+qR)^2 : (Rp+rP)^2 : (Pq+pQ)^2 )

Best regards
Nikos Dergiades