## 22486Fwd: [EGML] 2 circles and 5 lines in a right triangle [2 Attachments]

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• Jul 2, 2014
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---------- Forwarded message ----------
From: Seiichi Kirikami
Date: Wed, Jul 2, 2014 at 5:16 AM
Subject: [EGML] 2 circles and 5 lines in a right triangle [2 Attachments]
To: "anopolis@yahoogroups.com" <anopolis@yahoogroups.com>

[Attachment(s) from skirikami_0623@... included below]

Dear friends,

[1] Given a triangle ABC right in A, its incircle(I), its circumcircle(O), and its excircles (Ia), (Ib), (Ic) with their centers I, O, Ia, Ib, Ic, denote:
S, Q, R= the contact points of (I) with BC, CA, AB respectively,
Aa, Ba, Ca=the contact points of (Ia) with BC, CA, AB respectively,
Ab, Bb, Cb=the contact points of (Ib) with BC, CA, AB respectively,
Ac, Bc, Cc=the contact points of (Ic) with BC, CA, AB respectively,
B’, C’=the midpoints of CA, AB respectively,
B’’, C’’=the intersections of B’C’ with (O) respectively.
1. 8 points B’’, Q, R, C’’, Ac, Ca, Ba, and Ab are on a circle (1-circle).
2. 8 points B’’, Bb, Aa, S, Cc, C’’, Bc, and Cb are on another circle (2-circle).
See the attached pictures.
This is an extension of Le Probleme de Toshio Seimiya in Quelques Theoremes Oublies, vol. 1 (2007), Geometrie by Jean-Louis Ayme.

[2] Given a triangle ABC right in A, its excircles (Ia), (Ib), (Ic) with their centers Ia, Ib, Ic, and its angle bisectors (A-bsc), (B-bsc), (C-bsc), denote:
B’, C’=the midpoints of CA, AB respectively,
Ka, Kb, Kc=the intersections farthest from the sides of ABC of (A-bsc), (B-bsc), (C-bsc) with (Ia), (Ib), (Ic) respectibvely,
Jb, Jc=the intersections nearest to the sides of ABC of (B-bsc), (C-bsc) with (Ib), (Ic),
Mb, Mc=the intersections farthest from BC of (Ia) with the angle bisector of CBCa, BCBa respectively,
Lb, Lc=the intersections nearest to BC of (Ia) with the angle bisector of CBCa, BCBa respectively.
1. Kb, B’, Lb are on a line (1-line),
2. Kc, C’, Lc are on a line (2-line),
3. Mc, C’, Jc are on a line (3-line),
4. Mb, B’, Jb are on a line (4-line),
P= the intersection of 1-line and 2-line,
Q=the intersection of 3-line and 4-line.
5.P, Q, Ka are on a line (5-line).
See the attached pictures.
This is the continuation of Anopolis message #817.

Best regards,
Seiichi.