## [EMHL] Re: Reflections (a sequence of points)

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• Dear Peter [APH] ... I think that the 1st (ie for n=1) trilinear perspector is (sin3A/sin4A ::), and, in general, the n-th one is: (sin [2^(n+1) - 1]A / sin
Message 1 of 8 , Aug 26, 2005
Dear Peter

[APH]
>>
>> Let ABC be a triangle, and AoBoCo the orthic triangle of ABC.
>>
>> A1 := (Reflection of BoCo in AoBo) /\ (Reflection of BoCo in AoCo).
>>
>> A2 := (Reflection of BoCo in A1Bo) /\ (Reflection of BoCo in A1Co)
>>
>> .......
>>
>> An := (Reflection of BoCo in An-1Bo) /\ (Reflection of BoCo in
>> An-1Co)
>> Similarly Bn,Cn
>>
>> Are the triangles ABC, AnBnCn perspective?

[PM]:
>Yes!. Quite where I don't yet know.

I think that the 1st (ie for n=1) trilinear perspector is

(sin3A/sin4A ::), and, in general, the n-th one is:

(sin [2^(n+1) - 1]A / sin [2^(n+1)]A ::)

Locus:

Let ABC be a triangle, P a point and A'B'C' the pedal
triangle of P.

A1 := (Reflection of B'C' in A'B') /\ (Reflection of B'C' in A'C').

Similarly B1,C1.

Which is the locus of P such that ABC, A1B1C1 are perspective ?

If my computations are correct, then

( ABC, A1B1C1 are perspective ) <==>

cos(2Cb + Ab)
------------------ * CYCLIC = 1
cos(2Bc + Ac)

where Ab = angle(PAB), Ac = angle(PAC) etc.

This leads to a 9th degree equation in terms of x,y,z,
where (x:y:z) are the trilinears of P.

(H obviously lies on the locus, since Ab = 90-B deg. etc)

Antreas

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