## intersections of incircle and Steiner inellipse

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• 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
Message 1 of 4 , May 31, 2012
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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?

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?

Randy Hutson
• ... 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),
Message 2 of 4 , Jun 1, 2012
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> From: rhutson2 <rhutson2@...>
>
> 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?
>
> Randy Hutson

--
Barry Wolk
• Dear friends, [Randy Hutson] ... [Barry Wolk] ... [Randy Hutson] ... If the isotomic conjugates of the persectors of two inconics are interior points of ABC
Message 3 of 4 , Jun 1, 2012
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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
• Thanks Barry and Nikos.  I love it when these problems have such elegant solutions. Best regards, Randy Hutson ... [Non-text portions of this message have
Message 4 of 4 , Jun 4, 2012
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Thanks Barry and Nikos.  I love it when these problems have such elegant solutions.

Best regards,
Randy Hutson

>________________________________
>To: Hyacinthos@yahoogroups.com
>Sent: Friday, June 1, 2012 5:33 PM
>Subject: Re: [EMHL] intersections of incircle and Steiner inellipse
>
>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
>
>
>
>------------------------------------
>