Seven circles problem-I

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Seven circles problem-I

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#1 h Sep 21, 2013, 10:59 AM • 1 Y

Y by Adventure10

Seven circles problem-I

Given circle $(O)$ . Construct three circles $(A); (B); (C)$ which are pairwisely external-tangent and all of them are internal-tangent to $(O)$ . Construct three circles $(A_1); (B_1); (C_1)$ which satifies: $(A_1)$ is external-tangent to $(B); (C)$ , different from $(A_1)$ and internal-tangent to $(O); (B_1); (C_1)$ are defined similarly.
Prove that $AA_1,BB_1,CC_1$ are concurrent

//cdn.artofproblemsolving.com/images/a17f20ea66fcd110a6a71cc83d4daa884208c5fe.png

//cdn.artofproblemsolving.com/images/a17f20ea66fcd110a6a71cc83d4daa884208c5fe.png

This post has been edited 2 times. Last edited by daothanhoai, Sep 24, 2013, 4:01 AM

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nsato

#2 h Sep 21, 2013, 4:02 PM • 2 Y

Y by Adventure10 and 1 other user

See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=127681 (post 7).

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#3 h Sep 21, 2013, 5:27 PM • 2 Y

Y by Adventure10, Mango247

nsato wrote:

See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=127681 (post 7).

Have many problem for it. But I can't proof. I Want you rewrite it by your English. My English is bad

You see: https://www.facebook.com/photo.php?fbid=205244629628236&set=o.96389292832&type=1&theater
//cdn.artofproblemsolving.com/images/84d9a3d046d9acf50ec1c33a95c2e556aae35f38.png

//cdn.artofproblemsolving.com/images/84d9a3d046d9acf50ec1c33a95c2e556aae35f38.png

3-Many circle in show figure.
Three (A),(B),(C)circle (red) are tangent together and tangent with (G) and (K).
Three leve 1 (M11),(M21),(M31)circle (blue)
(M11) is tangent (B) and (C) and (G) (or (K))
(M21) is tangent (A) and (C) and (G) (or (K))
(M31) is tangent (A) and (B) and (G) (or (K))

Three leve n circle (M1i,M2i,M3i)
(M1i) is tangent (B) and (C) and (M1(i-1))
(M2i) is tangent (A) and (C) and (M2(i-1))
(M3i) is tangent (A) and (B) and (M3(i-1))
Hi is circle through tangent point (Mki) and (Mk(i-1)) ; k=1,2,3
Mji tangent (A),(B),(C) at Aji(j=1,2,3,4,5,6)

Broblem 1-AM1i,BM2i,CM3i concurrent

Broblem 2-Aji(j=1,2,3,4,5,6) are concyclic, Oi circumcircle of (A1iA2iA3iA4iA5iA6i)

Broblem 3-Ki,Oi,Hi(i=1,2,3,4….) are conlinear

//cdn.artofproblemsolving.com/images/e094dfa5dcfc8c459377ac8c105f390d05cf3137.png

//cdn.artofproblemsolving.com/images/e094dfa5dcfc8c459377ac8c105f390d05cf3137.png

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