A construction of the inverse of circumcevian triangles

The concept of inverse triangles is introduced in $ETC$ in the preamble above X(42005), by Clark Kimberling and Peter Moses, where we can find the cases of cevian and anticevian triangles.

Circumcevian triangles and their inverses are treated in  the preamble just before X(43344). 

However a construction of these triangles is missing at the moment.

 PDF: A construction of the inverse of circumcevian triangles

Looking for a circle

Given triangle ABC and a point P, we construct a conic through some six points and we want to determine P such that the conic becomes a circle. 

PDF: Looking for a circle.

Points with distances to the vertices proportional u:v:w

Here is the solution to this problem, with a final question:

Problem: Given three positive real numbers u, v, w find the points P such that the distances from P to the vertices to the reference triangle are proportional to u:v:w.

Points with distances to the vertices proportional u:v:w

The Bicevian Conic of X2 and X8

Again, a problem by Tran Quang Hung in ADGEOM is the starting point of a little research:

The Bicevian Conic of X2 and X8

Updated: Added the locus of the perspectors in the cubic case.

Two families of circles through the Feuerbach point

Presentamos una generalización de un problema propuesto por Tran Quang Hung en ADGEOM: 

Sea $ABC$ un triángulo con circuncentro $O$ y excentros $I_a$, $I_b$, $I_c$. Las rectas $OI_a$, $OI_b$, $OI_c$ cortan a las rectas $BC$, $CA$, $AB$ en $A'$, $B'$, $C'$, respectivametne. Entonces la circunferencia circunscrita al triángulo $A'B'C'$ pasa por el punto de Feuerbach del triángulo  $ABC$.
 

Two families of circles through the Feuerbach point

 

A metric relationship between Fermat points and isodynamic points

Let $A_1BC$, $AB_1C$, $ABC_1$ be the equilateral triangles erected outwards the triangle $ABC$ and $A_2BC$, $AB_2C$, $ABC_2$  the equilateral triangles erected inwards the triangle $ABC$. The Fermat points $X_{13}$ and $X_{14}$ are the perspectors of $ABC$ and the triangles $A_1B_1C_1$ and $A_2B_2C_2$ respetively. We consider the equal distances  $d_{13}=AA=BB_1=CC_1$ and $d_{14}=AA_2=BB_2=CC_2$.

On the other hand, we consider the points $X_{15}$ and $X_{16}$, that are the isogonal conjugates of Fermat points and also are the only points that have equilateral pedal triangles. If we call $l_{15}$ and $l_{16}$ the sidelengths of the corresponding pedal triangles, then we have the formula:

$d_{13} l_{15} = 2 \Delta = d_{14} l_{16}$,

where $\Delta$ is the area of the triangle $ABC$.

A conic centered at the Euler line

Theorem. Given a triangle $ABC$, call $\Gamma$ the locus of points $P$ such that polar of $P$ with respect to the circumcircle is tangent to the nine point circle. Then we have:
1) $\Gamma$ is a conic whose center is $X_{26}$, the circumcenter of the tangential triangle.
2) $\Gamma$ is an ellipse, parabola o hyperbola if and only if the triangle is acute, rectangle or obtuse.
3) The diameter on the transverse axis of the conic is also a diameter of the circumcircle of tangential triangle. Therefore, the transverse axis of the conic is the Euler line of the triangle.
4) If  $\alpha$, $\beta$  are the lengths of the transverse and conjugate axes of the conic respectively, we have the relation

                                $\left(\frac{\alpha}{\beta}\right)^2=1-\frac{OH^2}{R^2}$

5) The foci of the conic are $O$ and $O'$, where $O'$ is the reflection of $O$ on the center $X_{26}$.

In fact, this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".

Here is the version for two circles:

$(A)$ and $(B)$ are circles
The line $AB$ intersect $(B)$ at $M$ and $N$
$M'$ and $N'$ are the inverses of $M$ and $N$ with respect to $(A)$
$J$ is the inverse of $A$ with respect to $(B)$
$O$ is the inverse of $J$ with respect to $(A)$
$A'$ is the reflection of $A$ on $O$
The locus points $P$ such that the polar of $P$ with respect to $(A)$ is tangent to $(B)$ is a conic with foci $A$ and $A'$ and diameter $M'N'$.