GEOMETRÍA DINÁMICA

REEDITADA POR LA WEB : http://users.math.uoc.gr

Propiedad de las tangentes del círculo

Considere un círculo c, una tangente a ella L ₀ en su punto C y un paralelo L ₁ a L _{0 que} no se interseca c. Desde un punto H en L _1, dibuje las tangentes a c que 
intersecan L ₀ en los puntos {K, L}. Entonces CL * CK es constante.

Esta es una configuración interesante y el problema en cuestión brinda la oportunidad de enumerar algunas de sus propiedades. 
(1) El círculo c 'a través del centro A del círculo c y los puntos {K, L} tienen su centro P en la línea AH. 
(2) Los otros puntos de intersección {Q, R} de las tangentes {HL, HK} con el círculo c 'definen la línea QR simétrica a L ₀ wr a la línea AH. 
(3) El cuadrilátero LRKQ inscrito en c 'es un trapecio isósceles. 
(4) Los puntos (A, M, O, H) = -1 definen una división armónica. Aquí M es el diametral de A y O la intersección de {L ₀ , QR}. 
(5) El punto M, definido anteriormente, es un excéntrico del triángulo ALK.
(6) Los puntos (A, N, C, F) = -1 definen también una división armónica, donde N es el otro punto de intersección con c 'de AC y F es la intersección de {AC, L ₁ }. 

(1) se ve dibujando primero las líneas mediales de los segmentos AL, AK que se encuentran en P y definen allí el circuncentro de ALK. Sus paralelos de {L, K} se 
encuentran respectivamente en la M diametral de A en el circuncírculo c 'de ALK. Dado que son ortogonales a las bisectrices del triángulo HLK, son bisectrices externas 
de sus ángulos y definen un excéntrico del triángulo HLK. 
(2), (3), (5) son consecuencias inmediatas. 
(4) se desprende de la forma estándar de construir el polar de un punto H con respecto a un círculo c '(ver Polar2.html ).
(6) se deduce de (4) y la paralelismo de las líneas {L ₀ , L ₁ , CO, NM}. 
La reclamación inicial es una consecuencia de (6). Esta afirmación también es equivalente a la ortogonalidad del círculo c 'al círculo con diámetro FC.

Circle tangent to line and circle

Let c be a circle simultaneously tangent to line a and cirlce b. Then the two lines {CA, CB} through the contact point
with the cirlce b passing through A and its diametral point define on circle b a diameter DE orthogonal to line a.

That DE is a diameter is clear, since angle ACB = ECD is a right one. That DE is also orthogonal to a follows by the
angle equalities DEC = DCF, CAB = FCB, DCF+FCB = π/2.

 2. Circles tangent to two circles (intersecting case)

Consider two intersecting circles {k₁, k₂}. The set of all circles tangent to these two is the disjoint union of two subsets. One is
called direct [Johnson, p.111] and the other set is called transverse. All circles of the direct system do not separate
the two given circles. Every circle of the transverse system separates the two given circles.
The figure below displays two circles {c, c'} from the direct and transverse system correspondingly. Circles {c, c'} have a common
contact point with k₁ at its point A. The other contact with k₂ are B (of c) and E (of c').
[1]  Line AB which is the line of contacts of c (direct) passes through the outer center S₁ of similitude of {k₁, k₂}.
[2]  Line AE which is the line of contacts of c'(transverse) passes through the inner center S₂ of similitude of {k₁, k₂}.
[3]  The radical axes of all pairs of circles from {k₁, k₂, c, c'} pass through a point R.
[4]  Lines {AB,AE} intersect circle k₂ at diametral points {C, D} such that CD is orthogonal to the tangent of k₁ at A.
[5]  The centers of the circles of the direct system generate a hyperbola with foci at the centers {O₁, O₂} of the two given circles
       and major axis |r₁-r₂|, the difference of the radii of the given circles.
[6]  The centers of the circles of the transverse system generate an ellipse with foci at the centers {O₁, O₂} of the two given
       circles and major axis equal to |r₁+r₂|.

[1, 2] is proved in section-6 of RadicalAxis.html . [3] is proved in section-4 of the same file. [4] is proved here in section-1 and
[5, 6] are trivial verifications of the definitions of central conic sections through their basic focal properties.

 3. Circles tangent to two circles (non-intersecting case, inner)

Next figure illustrates the same properties as those of the preceding section but now for the case of two non-intersecting circles
{k₁, k₂} of which one contains the other.

In this case both the direct and transverse system of tangent circles are ellipses. The direct system consists of ellipses c with foci
at {O₁, O₂} and great axis equal to r₁+r₂. The transverse system consists of ellipses with the same as before foci and great axis
|r₁-r₂|.

 4. Circles tangent to two circles (non-intersecting case, outer)

Next figure illustrates the same properties as those of the preceding sections but now for the case of two non-intersecting circles
{k₁, k₂} which are outside to each other.

 5. Circles tangent to circle and line (non-intersecting)

Next figure illustrates the same properties as those of the preceding sections but now for the case of a line k₁ and a non-intersecting
circle k₂. In this case the centers of the variable tangent circles describe correspondingly two parabolas with directrices parallel
to line k₁ and at distance r₂ on either side of it. The direct system consisting corresponding to the directrix which is separated
from k₂ by line k₁. Both parabolas have their focus at the center O₂ of k₂.

 6. Circles tangent to circle and line (intersecting)

Next figure illustrates the same properties as those of the preceding sections but now for the case of a line k₁ and an intersecting
circle k₂. In this case the centers of the variable tangent circles describe correspondingly two parabolas with directrices parallel
to line k₁ and at distance r₂ on either side of it. Both parabolas have their focus at the center O₂ of circle k₂. In this case also
the two systems are indistinguishable.