Orthocentroidal circle, orthotransversal, concurrent lines
by kosmonauten3114, May 30, 2025, 2:43 PM
Let 
be a scalene oblique triangle, and 
a point on the orthocentroidal circle of (
).
Prove that the orthotransversal of, trilinear polar of the polar conjugate (
-isoconjugate) of , Droz-Farny axis of are concurrent.
The definition of the Droz-Farny axis of with respect to is as follows:
For a point
, there exists a pair of orthogonal lines 
, 
through such that the midpoints of the 3 segments cut off by , from the sidelines of are collinear. The line through these 3 midpoints is the Droz-Farny axis of wrt .
be a scalene oblique triangle, and 
a point on the orthocentroidal circle of (
).Prove that the orthotransversal of
, trilinear polar of the polar conjugate (
-isoconjugate) of , Droz-Farny axis of are concurrent.The definition of the Droz-Farny axis of
with respect to is as follows:For a point
, there exists a pair of orthogonal lines 
, 
through such that the midpoints of the 3 segments cut off by , from the sidelines of are collinear. The line through these 3 midpoints is the Droz-Farny axis of wrt .Attachments:
.![\[P=\dfrac{a}{\sqrt{2bc+1}}+\dfrac{b}{\sqrt{2ca+1}}+\dfrac{c}{\sqrt{2ab+1}}.\]](http://latex.artofproblemsolving.com/1/f/3/1f39a54e4cf378bfd4e66d97949350b9b37dc1ec.png)
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide 
,
.
such that 
is the square of an integer.
such that for every 
is satisfied? Explain the answer.
such that 
coprime, 
and 
.
.
be the centroid of triangle 
and let 
be the foot of the perpendicular from 
to side 
.
intersects the circumcircle 
of triangle 
be the point where 
intersects the circle 
,
and 
,
,
be a non-isosceles triangle with incenter 
Let 
be the smaller circle through 
tangent to 
and 
(the addition of indices being mod 3). Let 
be the second point of intersection of 
and 
Prove that the circumcentres of the triangles 
are collinear.
of the acute-angled triangle 
is tangent to its side 
at a point 
.
be an altitude of triangle 
be the midpoint of the segment 
is the common point of the circle 
(distinct from 
are tangent to each other at the point 
September 2024
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