Three Circles – One Inscribed Hexagon

I have found this simple method for construction of six concyclic points on sidelines of a triangle on April 6, 2006 and posted in Hyacinthos Geometry Forum.

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Detail as following:

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Theorem:

If three circles concur at one Cevian point and their other three common points lie on three Cevian lines and these three circles cut triangle sidelines at six points then these six points are concyclic.

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Proof:

Suppose P is a point on plane of triangle ABC; (O_a), (O_b), (O_c) are three circles concur at P and has three other common points: A_o on Cevian AP, B_o on Cevian BP, C_o on Cevian CP. It means: (O_a) is circumcircle of PB_oC_o, (O_b) is circumcircle of PC_oA_o, (O_c) is circumcircle of PA_oB_o .Suppose (O_a), (O_b), (O_c) cut triangle sidelines BC, CA, AB at A₁, A₂, B₁, B₂, C₁, C₂ respectively. We proof these six points are concyclic.

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We use power of circle theorem:
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- Power of A wrt circles (O_b) and (O_c):
AC₁*AC₂ = AA_o*AP = AB₁*AB₂ therefore B₁, B₂, C₁, C₂ are concyclic on one circle, suppose it is circle (K) and (K) cuts BC at D and E.

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- Power of B wrt circle (K): BC₁*BC₂ = BD*BE
- Power of B wrt circles (O_c): BC₁*BC₂ = BB_o*BP
Therefore BD*BE = BB_o*BP so D, E, B_o, P are on circle say (K₁)

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- Power of C wrt circle (K): CB₁*CB₂ = CD*CE
- Power of C wrt circles (O_b): CB₁*CB₂ = CC_o*CP
Therefore CD*CE = CC_o*CP so D, E, C_o, P are on circle say (K₂)

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Two circles (K₁), (K₂) have three common points D, E, P therefore they are the same one, say (K’). This circle (K’) passes P, B_o, C_o so it is the same circle (O_a) = (K’) and D, E are intersections of (O_a) and sideline BC. It means D, E are A₁, A₂ and six points A₁, A₂, B₁, B₂, C₁, C₂ are concyclic on one circle (K).

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Remarks:

1. Generally three circles (O_a), (O_b), (O_c) may be do not cut sidelines of ABC but in each special case of P we can choose proper circles (O_a), (O_b), (O_c) such that they always cut sidelines of ABC. For example:

2. If P is orthocenter H, we can choose (O_a) as circle with center at midpoint of BC and pass through H. (O_b), (O_c) are similarly defined.
3. If P is incenter I, we can choose (O_a) as circle with center at touch point on BC and pass through incenter I. (O_b), (O_c) are similarly defined.

4. With each P, we can choose some variants of (O_a), (O_b), (O_c) and will get some different interesting results.