## Re: [EMHL] Proofs of Myakishev's theorems about squares and circumcenters

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• ... [rest snipped] These results generalize to using similar rectangles instead of squares. If the rectangles have sizes a:af, b:bf, and c:cf, then the
Message 1 of 3 , Jun 12, 2003
--- Darij Grinberg wrote:
> In the following, I am going to establish some results of
> Alexei Myakishev, Jean-Pierre Ehrmann and me in Hyacinthos
> messages #6338, #6339, #6340, #6341, #6344, #6345.
>
> Consider a triangle ABC and the squares BBaCaC, CCbAbA and
> AAcBcB constructed upon its sides BC, CA, AB.
>
> _.-'Cb
> _.-`' \
> _.-' \
> Ab \
> \ \ _Ca
> \ \ _.-� \
> \ C-' \
> \ _.-' \ \
> \ _.-` \ Ba
> \ _.-' \ _.-�
> A----------------B-'
> | |
> | |
> | |
> | |
> | |
> | |
> | |
> Ac---------------Bc
>
> Let BcBa meet CaCb at A', and analogously define B' and C'.
> Let XYZ be the antipedal triangle of the centroid G of
> triangle ABC. Call Oa the circumcenter of triangle BaCaA',
> and similarly define Ob and Oc.
>
> Then,
>
> (1) XA' is orthogonal to BC.
>
> (2) Triangles A'B'C' and XYZ are homothetic.
>
> (3) The homothetic center T is the centroid of both triangles.
>
> (4) This T is the reflection of the centroid G of ABC in the
> circumcenter O of ABC.
>
> (5) The symmedian point T' of triangle A'B'C' lies on the
> Euler line of triangle ABC.
>
> (6) Triangles A'B'C' and OaObOc are homothetic.
>
> (7) The homothetic center is T'.
>
> PROOFS (entirely synthetic).

[rest snipped]

These results generalize to using similar rectangles instead of
squares. If the rectangles have sizes a:af, b:bf, and c:cf, then
the coordinates of all your points are functions of f, and all 7
of your results still hold in this general case. Do yuor proofs
generalize?
--
Barry Wolk

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• Dear Barry Wolk, ... Many thanks for this remark. The proofs do generalize! In fact, the crucial result that a median of triangle ABC is an altitude of the
Message 2 of 3 , Jun 12, 2003
Dear Barry Wolk,

You wrote:

>> These results generalize to using similar rectangles
>> instead of squares. If the rectangles have sizes a:af,
>> b:bf, and c:cf, then the coordinates of all your
>> points are functions of f, and all 7 of your results
>> still hold in this general case. Do yuor proofs
>> generalize?

Many thanks for this remark. The proofs do generalize!
In fact, the crucial result that a median of triangle
ABC is an altitude of the corresponding flank is true
for arbitrary similar rectangles.

Sincerely,
Darij Grinberg
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