Generalized Apollonian Circles

Journal for Geometry and Graphics Volume VOL (YEAR), No. NO, 1–5.

Paul Yiu

Department of Mathematical Sciences Florida Atlantic University Boca Raton, Florida, 33431, USA Email:[email protected]

Abstract. Given a triangle, we construct a one-parameter family of triads of coaxal circles orthogonal to circumcircle and mutually tangent to each other at a point on the circumcircle. This is a generalization of the classical Apollonian circles which intersect at the two isodynamic points.

The classical Apollonian circles of a triangle are the three circles each with diameter the segment between the feet of the two bisectors of an angle on its opposite sides. It is well known that the centers of the three circles are the intercepts of the Lemoine axis on the sidelines, 1 and that they have two points in common, the isodynamic points. The feet of the internal angle bisectors are the traces of the incenter. Those of the external angle bisectors are the intercepts of the trilinear polar of the incenter. More generally, given a point P , with traces X, Y , Z on the sidelines BC, CA, AB of triangle ABC, the harmonic conjugates of X in BC, Y in CA, Z in AB are the intercepts X, Y , Z of the trilinear polar of P on these sidelines. The circles with diameters XX, YY, ZZ are the Apollonian circles associated with P . The isodynamic points are the common points of the triad of Apollonian circles associated with the incenter I. In this note we characterize the points P for which these three Apollonian circles are mutually tangent to each other, and provide an elegant euclidean construction for such circles.

1The Lemoine axis of a triangle is the trilinear polar of the Lemoine symmedian point.

ISSN 1433-8157/$ 2.50 c YEAR Heldermann Verlag, Berlin 2 Paul Yiu: Generalized Apollonian circles We work with barycentric coordinates with respect to triangle ABC,and their homogenizations. See, for example, [3]. If P has homogeneous barycentric coordinates (u : v : w), its traces are the points X =(0:v : w),Y=(u :0:w),Z=(u : v :0). The harmonic conjugates of these points on the respective sidelines are X =(0:v : −w),Y =(−u :0:w),Z =(u : −v :0). These are the intercepts of the line x y z , u + v + w =0 the trilinear polar of P . The centers of the Apollonian circles, being the midpoints of the segments XX, YY, ZZ respectively, are the points (0 : v2 : −w2), (−u2 :0:w2), (u2 : −v2 :0). These are the intercepts of the line L: x y z + + =0, u2 v2 w2 the trilinear polar of the point (u2 : v2 : w2), which we call the barycentric square of P . See [3]. The circle Ca with diameter XX is the locus of points M satisfying

|BM| |CM| |BX| |CX| |BX| |CX| |v| |u| 1 1 . : = : = : = : = |u| : |v|

It follows that if M is a common point of Ca and Cb,then

|AM| |BM| |CM| 1 1 1 , : : = |u| : |v| : |w| and it is also a point of Cc. This means that the triad of circles are coaxal. Since X, X divide BC harmonically, so do B, C divide XX. It follows that B and C are inversive with respect to Ca, and any circle through them, in particular the circumcircle of ABC, is orthogonal to Ca. Similarly, the circumcircle C is orthogonal to Cb and Cc. This triad of circles generates a pencil of circles orthogonal to C. The radical axis contains the circumcenter O of triangle ABC, and is therefore the perpendicular from O to the line C of centers of the triad. Paul Yiu: Generalized Apollonian circles 3

Figure 1

Let Q bethepoleofthelineL with respect to the circumcircle C.This 2 means     c2 b2 1 0 u2  c2 a2  Q  1  ,  0  =  v2  b2 a2 1 0 w2 and b2 c2 a2 c2 a2 b2 a2 b2 c2 Q a2 − b2 − c2 − . = v2 + w2 u2 : w2 + u2 v2 : u2 + v2 w2 (1)

This has a very simple geometric meaning. It is the perspector of the cevian triangle of (u2 : v2 : w2) and the anticevian triangle of (a2 : b2 : c2), which is the tangential triangle.

Theorem 1

The Apollonian circles of P intersect at real points if and only if the point Q lies inside or on the circumcircle. Proof. The real intersections lie on the radical axis, which passes through O. Their midpoint is the intersection of the radical axis and L,withinversive image Q in the circumcircle. Since the circles of the triad are orthogonal to C, these intersections are inversive with respect to C, and their midpoint lies outside or on C. It follows that Q lies inside or on C. Conversely, ... 2

In particular, the circles in the triad are mutually tangent to each other if and only if Q lies on the circumcircle. In this case, the common point is Q. The equation of the circumcircle is

a2yz + b2zx + c2xy =0. (2)

In fact, for a point P with homogeneous barycentric coordinates (x : y : z) lies inside or on C if and only a2yz+ b2zx+ c2xy ≥ 0. 3 Similarly, the equation

yz + zx + xy =0

2Here is an abuse of notations in which we treat Q as a 1 × 3 matrix. In the next step, we reinterpret Q as homogeneous coordinates. 3For a ﬁnite point P =(x : y : z), the power of P with respect to C is given by a2yz+b2zx+c2xy (x+y+z)2 . This is the product XP · PY of signed lengths for an arbitrary line through P intersecting the circumcircle at X and Y . 4 Paul Yiu: Generalized Apollonian circles represents the Steiner circum-ellipse E (with center at the centroid G of triangle ABC). A point P is inside or on E if and only if

yz + zx + xy ≥ 0.

From the coordinates of Q given in (1), it is clear that Q lies inside or on C if and only if the point b2 c2 a2 c2 a2 b2 a2 b2 c2 Q = + − : + − : + − v2 w2 u2 w2 u2 v2 u2 v2 w2 lies inside or on E, i.e.,

≤ c2 a2 − b2 a2 b2 − c2 a2 b2 − c2 b2 c2 − a2 0 w2 + u2 v2 u2 + v2 w2 + u2 + v2 w2 v2 + w2 u2 b2 c2 − a2 c2 a2 − b2 + v2 + w2 u2 w2 + u2 v2 2 2 2 2 2 2 4 4 4 b c c a a b − a − b − c =2 v w +2 w u +2 u v u v w a b c − a b c a − b c a b − c . = u + v + w u + v + w u v + w u + v w

Corollary 2

Given lengths da, db, dc, there are two points M satisfying AM : BM : CM = da : db : dc if and only if there is a triangle of side lengths ada, bdb and cdc. If this condition is satisﬁed, the two points M are the intersections of the Apollonian circles of the point P = 1 : 1 : 1 . The two points coincide (on da db dc the circumcircle) if such a triangle degenerates into a line segment.

Theorem 3

The following statements are equivalent. (a) The Apollonian circles of P are tangent to each other. (b) The point Q lies on the circumcircle.

Figure 2 Paul Yiu: Generalized Apollonian circles 5

The circum-ellipse Cm has center at the Mittenpunkt with coordinates (a(b + c − a):b(c + a − b):c(a + b − c)). 4 This is the intersection of the three lines each joining an excenter to the midpoint of the corresponding side 5 of ABC. The circum-ellipse Cm intersects the circumcircle C at the point a b c T . = b − c : c − a : a − b

If a line through T intersects the circumcircle at Q and the circumconic Cm at P , then the Apollonian circles associated with P are mutually tangent at Q. This leads to a remarkable animation picture showing a moving triad of circles orthogonal to the circumcircle, mutually tangent to each other at a point on C. As an interesting special case, if P is the intersection S of Em with the Steiner circum-ellipse E, then the Apollonian circles are tangent to each other at T . See Figure 3. Furthermore, the line of centers is tangent to E at S and contains T . 6

Figure 3

References

[1] C. Kimberling, Triangle centers and central triangles, Congressus Numer- antium, 129 (1998) 1–285. [2] C. Kimberling, Encyclopedia of Triangle Centers, available at http://faculty.evansville.edu/ck6/encyclopedia/ETC.html. [3] P. Yiu, The uses of homogeneous barycentric coordinates in plane euclidean geometry, Int. J. Math. Educ. Sci. Technol., 31 (2000) 569–578. Received

4 This is the point X9 in [2]. 5 This is X100 in [2]. It can be constructed as the point of tangency of the circumcircle and the incircle of the superior triangle, the one bounded by the parallels to the sides of ABC through the opposite vertices. 6This is the line (b − c)2x +(c − a)2y +(a − b)2z =0.