Re: Radical Axes and Loci
- Dear Antreas and other Hyacinthists
> For any three circles (A,r1), (B,r2), (C,r3) centered on A,B,C the radicalaxes
> of (A,r1), (B,r2)), ((B,r2), (C,r3)), ((C,r3),(A,r1)) are obviouslyconcurrent.
>Some easy remarks :
> So we can always ask the same questions for loci.
> But the problem is to define interesting radii of the circles!!
- (O) is the circumcircle, (A) is the circle (A,r1),..
- UVW is the triangle bounded by the radical axis of the circles
- O' is the circumcenter of UVW
- D is the radical center of (A),(B),(C)
1) DO' is parallel to the Euler line of ABC
Explanation : the lines VW,WU,UV are perpendicular respectively to OA,OB,OC;
thus UVW and the orthic triangle A'B'C' are homothetic and the homothecy h maps
the 9P-center N to O'.
As U is the radical center of (O),(B),(C), DU&BC are perpendicular; so do DV&CA, DW&AB. Hence D = h(H) and the result
We can notice too that D is the incenter (or an excenter if ABC is obtusangle)
of UVW and that O & D are the orthologic centers of ABC & UVW
2) The locus of the points M for which
(b^2-c^2)AM^2+(c^2-a^2)BM^2+(a^2-b^2)CM^2=0 is a line through O,G,H; so it is
the Euler line of ABC
As AD^2-r1^2 = BD^2-r2^2 = CD^2-r3^2, we have
It follows from 1) and 2) that we have 3 equivalent conditions :
a) O' lies on the Euler line of ABC
b) D lies on the Euler line of ABC
c) (b^2-c^2)r1^2+(c^2-a^2)r2^2+(a^2-b^2)r3^2 = 0
So, we can consider that c) is an interesting condition for the radii