viernes, 26 de octubre de 2012

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 $M'N'$.


miércoles, 24 de octubre de 2012

A property of the orthocentroidal circle

If $ABC$ is a triangle, the circle with $GH$ as diameter, where $G$ and $H$ are the centroid and orthocenter of $ABC$, is called the orthocentroidal circle of $ABC$.
 
We present the following property of the orthocentroidal circle of a triangle: The locus of points $P$ such that the trilinear polar of $P$ goes through the inverse of $P$ on the circumcircle is a quartic, the isogonal conjugate of the orthocentroidal circle.

In the following figure, $Q$ lies on the orthocentroidal circle, $P$ is its isogonal conjugate and $P'$ is the inverse of $P$ on the circumcircle. $A'B'C'$ is the cevian triangle of $P$ and p is the trilinear polar of $P$, through $P'$.





viernes, 5 de octubre de 2012

Los números vampiros

En la página Didáctica Especializada encontramos el siguiente acertijo:

Los números vampiros son números que cumplen estas reglas:
  • Tienen un número par de cifras.
  • Las cifras se pueden reordenar para formar dos números igual de largos, que multiplicados dan el número original.
Los dos números cortos se llaman los colmillos del vampiro y no pueden terminar los dos en 0.
Por ejemplo, 1435 es un número vampiro y sus colmillos son 35 y 41. ¿Por qué? Pues porque 35×41=1435.

Para resolver el problema con Mathematica, comenzamos por definir una función que producirá un texto del tipo 35x41 = 1435 cuando encontremos esa solución:
TextoSalida[{m_, n_}] :=
 ToString[m] <> " x " <> ToString[n] <> " = " <>  ToString[m n] 

A continuación definimos una función que obtendrá todos los números vampiros entre un valor inicial y un valor final: BuscarVampiros[inicio_, final_] := Module[ {lista, n, digits, factors, sol}, lista={}; For[n=inicio, n < final, n++, digits=IntegerDigits[n]; factors=Map[ Map[FromDigits,#]&, Map[Partition[#,{Length[digits] / 2}]&, Permutations[digits]]]; sols=Select[factors,Apply[Times, #]==n&]; If[sols!={},lista=Append[lista,sols[[1]]]]; ]; Map[TextoSalida,lista] // ColumnForm ] Ahora podemos buscar los números vampiros de cuatro cifras: BuscarVampiros[1000,9999] 21 x 60 = 1260 15 x 93 = 1395 41 x 35 = 1435 51 x 30 = 1530 87 x 21 = 1827 27 x 81 = 2187 86 x 80 = 6880