--- 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

__________________________________

Do you Yahoo!?

Yahoo! Calendar - Free online calendar with sync to Outlook(TM).

http://calendar.yahoo.com