HYACINTHOS 16782

 

--- Antreas P. Hatzipolakis wrote:

>
> Let ABC be a triangle, L a line passing through P and Q,
> and La,Lb,Lc the parallels to L through A,B,C, resp.
>
> Let Ma,Mb,Mc be the Reflections of BC,CA,AB, in
> La,Lb,Lc, resp.
>
> What can be said about the triangle bounded by the
> lines (Ma,Mb,Mc)?
>
> If Q is fixed point, which is the locus of P such that
> ABC, (MaMbMc) are perspective?
>
> If P'Q' is the PQ-line of (Ma,Mb,Mc) are the lines
> PQ, P'Q' parallels?
> (Definition: If P = (x:y:z) wrt triangle ABC, then
> P' = (x:y:z) wrt the triangle (Ma,Mb,Mc))
>
> What is the intersection point of PP' and QQ'?
>
> Special Cases:
>
> PQ = OH line (Euler Line) or OI line of ABC.
>
>
> APH
>

[Randy Hutson]:

The triangle bounded by (MaMbMc) is similar to, and 3 times the size
of, ABC. If this triangle is reflected about the original line, the
resultant triangle is homothetic to ABC. For the Euler Line, the
homothetic center is X(125) (center of Jerabek Hyperbola). For the
line OI, the homothetic center id X(11) (Feuerbach Point). For line
IG, the homothetic center is not found in ETC (search value
-0.928012514996018). For line X(11)X(244) (Feuerbach Tangent Line,
i.e. line tangent to incircle and nine-point circle at Feuerbach
Point), the homothetic center is not found in ETC (search value
-1.466654575213205).