The Apollonius Circles of rank đ?’Œ Professor Ion Patrascu, Fratii Buzesti National College, Craiova, Romania Professor Florentin Smarandache, New Mexico University, Gallup, NM, USA
The purpose of this article is to introduce the notion of Apollonius circle of rank k and generalize some results on Apollonius circles.
Definition 1. It is called an internal cevian of rank k the line đ??´đ??´đ?‘˜ where đ??´đ?‘˜ ∈ (đ??ľđ??ś), such that
đ??ľđ??´ đ??´đ?‘˜ đ??ś
đ??´đ??ľ đ?‘˜
= ( ) (đ?‘˜ ∈ â„?). đ??´đ??ś
If đ??´â€˛đ?‘˜ is the harmonic conjugate of the point đ??´đ?‘˜ in relation to đ??ľ and đ??ś, we call the line đ??´đ??´â€˛đ?‘˜ an outside cevian of rank k.
Definition 2. We call Apollonius circle of rank k with respect to the side đ??ľđ??ś of đ??´đ??ľđ??ś triangle the circle which has as diameter the segment line đ??´đ?‘˜ đ??´â€˛đ?‘˜ .
Theorem 1. Apollonius circle of rank k is the locus of points đ?‘€ from đ??´đ??ľđ??ś triangle's plan, satisfying the relation:
đ?‘€đ??ľ đ?‘€đ??ś
đ??´đ??ľ đ?‘˜
=( ) . đ??´đ??ś
Proof. Let đ?‘‚đ??´đ?‘˜ the center of the Apollonius circle of rank k relative to the side đ??ľđ??ś of đ??´đ??ľđ??ś triangle (see Figure 1) and đ?‘ˆ, đ?‘‰ the points of intersection of this circle with the circle circumscribed to the triangle đ??´đ??ľđ??ś. We denote by đ??ˇ the middle of arc đ??ľđ??ś, and we extend đ??ˇđ??´đ?‘˜ to intersect the circle circumscribed in đ?‘ˆ ′ . In đ??ľđ?‘ˆ ′ đ??ś triangle, đ?‘ˆ ′ đ??ˇ is bisector; it follows that
đ??ľđ??´đ?‘˜ đ??´đ?‘˜ đ??ś
=
đ?‘ˆâ€˛đ??ľ đ?‘ˆâ€˛đ??ś
đ??´đ??ľ đ?‘˜
= ( ) , so đ?‘ˆ ′ đ??´đ??ś
belongs to the locus. The perpendicular in đ?‘ˆ ′ on đ?‘ˆ ′ đ??´đ?‘˜ intersects BC on đ??´â€˛â€˛ đ?‘˜, which is the foot of the đ??ľđ?‘ˆđ??ś triangle's outer bisector, so the harmonic conjugate
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′ ′ of đ??´đ?‘˜ in relation to đ??ľ and đ??ś, thus đ??´â€˛â€˛ đ?‘˜ = đ??´đ?‘˜ . Therefore, đ?‘ˆ is on the Apollonius
circle of rank đ?‘˜ relative to the side đ??ľđ??ś, hence đ?‘ˆ ′ = đ?‘ˆ.
Figure 1
Let đ?‘€ a point that satisfies the relation from the statement; thus đ??ľđ??´đ?‘˜ đ??´đ?‘˜ đ??ś
đ?‘€đ??ľ đ?‘€đ??ś
=
; it follows – by using the reciprocal of bisector's theorem – that đ?‘€đ??´đ?‘˜ is the
internal bisector of angle đ??ľđ?‘€đ??ś. Now let us proceed as before, taking the external bisector; it follows that đ?‘€ belongs to the Apollonius circle of center đ?‘‚đ??´đ?‘˜ . We consider now a point đ?‘€ on this circle, and we construct đ??ś ′ such that âˆ˘đ??ľđ?‘ đ??´đ?‘˜ ≥ ̂′ ). Because đ??´â€˛đ?‘˜ đ?‘ ⊼ âˆ˘đ??´đ?‘˜ đ?‘ đ??ś ′ (thus (đ?‘ đ??´đ?‘˜ is the internal bisector of the angle đ??ľđ?‘ đ??ś đ?‘ đ??´đ?‘˜ , it follows that đ??´đ?‘˜ and đ??´â€˛đ?‘˜ are harmonically conjugated with respect to đ??ľ and đ??ś ′ . On the other hand, the same points are harmonically conjugated with respect to đ??ľ and đ??ś; from here, it follows that đ??ś ′ = đ??ś, and we have đ??´đ??ľ đ?‘˜
(đ??´đ??ś ) .
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đ?‘ đ??ľ đ?‘ đ??ś
=
đ??ľđ??´đ?‘˜ đ??´đ?‘˜ đ??ś
=
Definition 3. It is called a complete quadrilateral the geometric figure obtained from a convex quadrilateral by extending the opposite sides until they intersect. A complete quadrilateral has 6 vertices, 4 sides and 3 diagonals.
Theorem 2. In a complete quadrilateral, the three diagonals' means are collinear (Gauss - 1810).
Proof. Let đ??´đ??ľđ??śđ??ˇđ??¸đ??š a given complete quadrilateral (see Figure 2). We denote by đ??ť1 , đ??ť2 , đ??ť3 , đ??ť4 respectively the orthocenters of đ??´đ??ľđ??š, đ??´đ??ˇđ??¸, đ??śđ??ľđ??¸, đ??śđ??ˇđ??š triangles, and let đ??´1 , đ??ľ1 , đ??š1 the feet of the heights of đ??´đ??ľđ??š triangle.
Figure 2
As previously shown, the following relations occur: đ??ť1 đ??´. đ??ť1 đ??´1 − đ??ť1 đ??ľ. đ??ť1 đ??ľ1 = đ??ť1 đ??š. đ??ť1 đ??š1 ; they express that the point đ??ť1 has equal powers to the circles of diameters đ??´đ??ś, đ??ľđ??ˇ, đ??¸đ??š , because those circles contain respectively the points đ??´1 , đ??ľ1 , đ??š1 , and đ??ť1 is an internal point. It is shown analogously that the points đ??ť2 , đ??ť3 , đ??ť4 have equal powers to the same circles, so those points are situated on the radical axis (common to the circles), therefore the circles are part of a 3
fascicle, as such their centers – which are the means of the complete quadrilateral's diagonals – are collinear. The line formed by the means of a complete quadrilateral's diagonals is called Gauss line or Gauss-Newton line.
Theorem 3. The Apollonius circles of rank k of a triangle are part of a fascicle.
Proof. Let đ??´đ??´đ?‘˜ , đ??ľđ??ľđ?‘˜ , đ??śđ??śđ?‘˜ be concurrent cevians of rank k and đ??´đ??´â€˛đ?‘˜ , đ??ľđ??ľđ?‘˜â€˛ , đ??śđ??śđ?‘˜â€˛ be the external cevians of rank k (see Figure 3). The figure đ??ľđ?‘˜â€˛ đ??śđ?‘˜ đ??ľđ?‘˜ đ??śđ?‘˜â€˛ đ??´đ?‘˜ đ??´â€˛đ?‘˜ is a complete quadrilateral and Theorem 2 is applied.
Figure 3
Theorem 4. The Apollonius circles of rank k of a triangle are the orthogonals of the circle circumscribed to the triangle.
Proof. We unite đ?‘‚ to đ??ˇ and đ?‘ˆ (see Figure 1), đ?‘‚đ??ˇ ⊼ đ??ľđ??ś and ′ ′ 0 Ě‚ Ě‚ Ě‚ đ?‘š(đ??´Ě‚ đ?‘˜ đ?‘ˆđ??´đ?‘˜ ) = 90 , it follows that đ?‘ˆđ??´đ?‘˜ đ??´đ?‘˜ = đ?‘‚đ??ˇđ??´đ?‘˜ = đ?‘‚đ?‘ˆđ??´đ?‘˜ . The congruence ′ Ě‚ Ě‚ đ?‘ˆđ??´ đ?‘˜ đ??´đ?‘˜ ≥ đ?‘‚đ?‘ˆđ??´đ?‘˜ shows that đ?‘‚đ?‘ˆ is tangent to the Apollonius circle of center đ?‘‚đ??´đ?‘˜ .
Analogously, it can be demonstrated for the other Apollonius circles.
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Remark 1. The previous Theorem indicates that the radical axis of Apollonius circles of rank k is the perpendicular taken from đ?‘‚ to the line đ?‘‚đ??´đ?‘˜ đ?‘‚đ??ľđ?‘˜ .
Theorem 5. The centers of Apollonius circles of rank k of a triangle are situated on the trilinear polar associated to the intersection point of the cevians of rank 2k.
Proof. From the previous Theorem, it results that đ?‘‚đ?‘ˆ ⊼ đ?‘ˆđ?‘‚đ??´đ?‘˜ , so đ?‘ˆđ?‘‚đ??´đ?‘˜ is an external cevian of rank 2 for đ??ľđ??śđ?‘ˆ triangle, thus an external symmedian. Henceforth,
đ?‘‚ đ??´đ?‘˜ đ??ľ đ?‘‚đ??´ đ?‘˜ đ??ś
đ??ľđ?‘ˆ 2
=(
đ??´đ??ľ 2đ?‘˜
) = (đ??´đ??ś ) đ??śđ?‘ˆ
(the last equality occurs because đ?‘ˆ belong
to the Apollonius circle of rank đ?‘˜ associated to the vertex đ??´).
Theorem 6. The Apollonius circles of rank k of a triangle intersect the circle circumscribed to the triangle in two points that belong to the internal and external cevians of rank đ?‘˜ + 1.
Proof. Let đ?‘ˆ and đ?‘‰ points of intersection of the Apollonius circle of center đ?‘‚đ??´đ?‘˜ with the circle circumscribed to the đ??´đ??ľđ??ś (see Figure 1). We take from đ?‘ˆ and đ?‘‰ the perpendiculars đ?‘ˆđ?‘ˆ1 , đ?‘ˆđ?‘ˆ2 and đ?‘‰đ?‘‰1 , đ?‘‰đ?‘‰2 on đ??´đ??ľ and đ??´đ??ś respectively. The quadrilaterals đ??´đ??ľđ?‘‰đ??ś , đ??´đ??ľđ??śđ?‘ˆ are inscribed, it follows the similarity of triangles đ??ľđ?‘‰đ?‘‰1 , đ??śđ?‘‰đ?‘‰2 and đ??ľđ?‘ˆđ?‘ˆ1 , đ??śđ?‘ˆđ?‘ˆ2 , from where we get the relations: đ??ľđ?‘‰ đ?‘‰đ?‘‰1 = , đ??śđ?‘‰ đ?‘‰đ?‘‰2
But
đ??ľđ?‘‰ đ??śđ?‘‰
đ??´đ??ľ đ?‘˜ đ?‘ˆđ??ľ
=( ) , = đ??´đ??ś đ?‘ˆđ??ś
đ?‘˜ đ??´đ??ľ đ?‘˜ đ?‘‰đ?‘‰1 ) (đ??´đ??ś ) , đ?‘‰đ?‘‰ = (đ??´đ??ľ đ??´đ??ś 2
đ?‘ˆđ??ľ đ?‘ˆđ?‘ˆ1 = . đ?‘ˆđ??ś đ?‘ˆđ?‘ˆ2 đ?‘˜
đ??´đ??ľ 1 and đ?‘ˆđ?‘ˆ = (đ??´đ??ś) , relations that show that đ?‘ˆđ?‘ˆ 2
đ?‘‰ and đ?‘ˆ belong respectively to the internal cevian and the external cevian of rank đ?‘˜ + 1. 5
Definition 4. If the Apollonius circles of rank k associated with a triangle have two common points, then we call these points isodynamic points of rank đ?‘˜ (and we denote them đ?‘Šđ?‘˜ , đ?‘Šđ?‘˜â€˛ ).
Property 1. If đ?‘Šđ?‘˜ , đ?‘Šđ?‘˜â€˛ are isodynamic centers of rank k, then: đ?‘Šđ?‘˜ đ??´. đ??ľđ??ś đ?‘˜ = đ?‘Šđ?‘˜ đ??ľ. đ??´đ??ś đ?‘˜ = đ?‘Šđ?‘˜ đ??ś. đ??´đ??ľđ?‘˜ ; đ?‘Šđ?‘˜â€˛ đ??´. đ??ľđ??ś đ?‘˜ = đ?‘Šđ?‘˜â€˛ đ??ľ. đ??´đ??ś đ?‘˜ = đ?‘Šđ?‘˜â€˛ đ??ś. đ??´đ??ľđ?‘˜ . The proof of this property follows immediately from Theorem 1.
Remark 2. The Apollonius circles of rank 1 are the investigated Apollonius circles (the bisectors are cevians of rank 1). If đ?‘˜ = 2, the internal cevians of rank 2 are the symmedians, and the external cevians of rank 2 are the external symmedians, i.e. the tangents in triangle’s vertices to the circumscribed circle. In this case, for the Apollonius circles of rank 2, Theorem 3 becomes:
Theorem 7. The Apollonius circles of rank 2 intersect the circumscribed circle to the triangle in two points belonging respectively to the antibisector's isogonal and to the cevian outside of it.
The proof follows from the Proof of Theorem 6. We mention that the antibisector is isotomic to the bisector, and a cevian of rank 3 is isogonic to the antibisector.
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