PORISTIC TRIANGLES OF THE ARBELOS

Transcription

1 INTERNTINL JURNL F EMETRY Vol. 6 (2017), No. 1, PRISTIC TRINLES F THE RELS ntonio utierrez, Hiroshi kumura and Hussein Tahir bstract. The arbelos is associated with poristic triangles whose circumcircle is the outer circle forming the arbelos. The triangles give infinitely many sextuplets of rchimedean circles and their intouch triangles share the same nine-point circle, which is also rchimedean and has center on the Schoch line. 1. Introduction n arbelos is one of the two congruent areas surrounded by three mutually touching circles with collinear centers in the plane in a restricted sense. Circles having common radius equal to the half the harmonic mean of the radii of the two inner circles are said to be rchimedean, which are one of the main topics on the arbelos. In this article, we show that the arbelos can naturally be associated with a set of poristic triangles, whose circumcircle is the outer circle forming the arbelos. The triangles give two families of infinitely many sextuplets of rchimedean circles. The intouch triangles of the poristic triangles share the same nine-point circle, which is also rchimedean and has center at the point of intersection of the Schoch line and the line passing through the centers of the circles forming the arbelos. Some special cases are considered. 2. ase triangle of the arbelos In this section we construct a special triangle of the arbelos. For two points P and Q, P (Q) denotes the circle with center P passing through Q, and (P Q) denotes the circle with a diameter P Q. The center of a circle δ is denoted by δ. Let be a point on the segment, and let = (), = () and = (). The configuration of the three circles is denoted by (,, ) and called an arbelos. Let a = /2 and b = /2. Circles of radius ab/(a+ b) are called rchimedean circles of (,, ), or said to be rchimedean with Keywords and phrases: arbelos, rchimedean circles, nine-point circle, Schoch line, poristic triangles (2010)Mathematics Subject Classification: 51M04, 51N20 Received: In revised form: ccepted:

2 28 ntonio utierrez, Hiroshi kumura and Hussein Tahir respect to (,, ), and the common radius is denoted by r. We use a rectangular coordinate system with origin such that the points and have coordinates (2a, 0) and ( 2b, 0) respectively. The radical axis of and is called the axis of the arbelos, which overlaps with the y-axis. The point of intersection of the ( axis and lying in the region y > 0 is denote by I. It has coordinates 0, 2 ) ab. The external common tangent of and touching the two circles in the region y < 0 is expressed by the equation [8]: (1) (a b)x + 2 aby + 2ab = 0. I K J Figure 1. Let J and K be the points of intersection of and this tangent, where J is closer to than K (see Figure 1). We call IJK the base triangle of (,, ). Let g = a + b, u = a b, w = a 2 + b 2 and v = w + ab. The points J and K have coordinates ( 2r (u 2v), 2 ab(w uv) g g 2 ) and ( ) 2r (u + 2v), 2 ab(w + uv) g g 2, respectively. Therefore the lines IJ and KI are expressed by the equations (2) vx aby + 2ab = 0 and vx aby + 2ab = 0, respectively. The equations show that the lines IJ and KI are symmetric in the axis. They also show that each of the distances from to IJ and from to KI is 2ab/ v 2 + ab = 2r. n the other hand, the distance between and JK also equals 2r [2], which is also obtained from (1). Therefore we get: Theorem 2.1. The base triangle IJK has incircle of radius 2r with center. 3. Poristic triangles of the arbelos Let be the incircle of the triangle IJK. Since IJK has circumcircle and incircle, there is a continuous family of triangles with the same circumcircle and incircle by the Poncelet closure theorem. We call the triangles the poristic triangles of (,, ). In this section we show that each of the poristic triangles of (,, ) gives several rchimedean circles and the intouch triangles of the poristic triangles share the same nine-point circle.

3 Poristic triangles of the arbelos 29 Let EF be a poristic triangle of (,, ), and let E F be its intouch triangle, where E, F and lie on the segments F, E and EF, respectively (see Figure 2). Let E m be the midpoint of E. The points F m and m are defined similarly. Theorem 3.1. The following statements hold. (i) The circles (E) and (F ) share the chord, and the circle ( ) is rchimedean and is the inverse of the line EF in the circle for the points E, F and. Similar facts are true for the points E, F and and for E, F and. (ii) The circle with center E m touching the sides E and EF is rchimedean. Similar facts are true for the points F m and m and the corresponding sides of the triangle EF. (iii) The points E m, F m and m lie on the circle ( ). Proof. The part (i) is obvious. The homothety with center E and ratio 1/2 carries the circle and the point into the rchimedean circle touching EF and E and the point E m, respectively. This proves (ii). The homothety with center and ratio 1/2 carries and E into the circle ( ) and E m. This proves (iii). E E m F m F m E F Figure 2. We get a sextuplet of rchimedean circles from a poristic triangle of (,, ) by the theorem. Therefore we get infinitely many sextuplets of rchimedean circles. For a triangle with inradius 2r, we can construct an arbelos with rchimedean circles of radius r and a poristic triangle. Corollary 3.2. For a triangle with inradius 2r and incenter, let be the circumcircle of. If the line intersects at points and, let = ( ) and = ( ). Then (,, ) is an arbelos with rchimedean circles of radius r and a poristic triangle. Let s = r u/g. The line expressed by the equation x = s is called the Schoch line of (,, ) [9]. Let E be the point of intersection of (F ) and

4 30 ntonio utierrez, Hiroshi kumura and Hussein Tahir ( ) different from. The points F and are defined similarly. Let ε be the circle passing through E, F and (see Figure 3). Then ε is rchimedean with respect to (,, ) [4]. Theorem 3.3. The following statements are true. (i) The point E is the midpoint of F and lies on the segment E. Similar facts are true for the points F and. (ii) The circle ε is the inverse of in the circle and is the nine-point circle of the triangle E F. The center of ε coincides with the point of intersection of the Schoch line and. Proof. The line F is the inverse of the circle (E) in, and E F is a kite. Hence E is the midpoint of F. This proves (i). Since E is the inverse of E in and similar facts are true for F and F and for and, the circle ε is the inverse of in. The inverses of and in are the endpoints of a diameter of ε and have x-coordinates 2r 2 /a and 2r2 /b. Hence the center of ε has x-coordinate r 2 /a r2 /b = s. The rest of (ii) is obvious. Corollary 3.4. The intouch triangles of the poristic triangles of (,, ) share the same rchimedean nine-point circle ε. In general, the intouch triangles of the poristic triangles with the same circumcircle and incircle of a triangle share the same nine-point circle. F E ε ε F W V E Figure 3. The theorem creates a new aspect of the Schoch line: it is the perpendicular to passing through the center of the inverse of in. If n = r /g, then ε coincides with the circle with center on the Schoch line and touching the two circles with centers (na, 0) and (nb, 0) passing through internally [9]. Let V (resp. W ) be the external center of similitude of the circles ε and (resp. ). Then the circle (V ) is rchimedean (see Figure 3). For V has coordinates (( r a + as)/(a r ), 0) = ( 2r, 0). Similarly (W ) is rchimedean. The circle (W ) (resp. (V )) is known as an rchimedean circle touching the axis from the side opposite to (resp. ) and the tangents from to (resp. to ), and is denoted by W 6 (resp. W 7 ) in

5 Poristic triangles of the arbelos 31 [2]. Therefore the fact creates a new aspect of the rchimedean circles W 6 and W n arbelos derived from Dao s result From the arbelos (,, ) we can construct another arbelos, which shares rchimedean circles with (,, ). Suppose that the external common tangent of and touching the two circles in the region y > 0 intersects at points S and T (see Figure 4). Let U be the point of intersection of the tangents of at S and T. Then SUT is a kite. Hence the circle ( U) passes through the points S and T. n the other hand, Dao Thanh ai has shown that each of the distances from the point I to the two tangents equals 2r [1]. While the distance between I and the line ST is also 2r [7]. Hence the triangle ST U has incircle of radius 2r with center I. Therefore if = ( I), = (UI) and = ( U), then the arbelos (,, ) shares rchimedean circles with (,, ), and ST U is a poristic triangle of (,, ) by Corollary 3.2. The axis of (,, ) is the tangent of at the point I, which is parallel to ST [7]. Let a and b be the radii of the circles and, respectively. Then a = g/2. Solving the equation a b /(a + b ) = r for b, we get b = abg/w. Therefore has radius g 3 /(2w). U T I S Figure nother infinitely many sextuplets of rchimedean circles We consider another infinitely many sextuplets of rchimedean circles of (,, ), which are obtained from the following simple fact (see Figure 5): Proposition 5.1. If D is a point lying outside a circle C and M is a point lying on one of the tangents of C from D, then the distance between M and the line D C equals the radius of C if and only if M lies on the circle D( C ). Proof. If N is the foot of perpendicular from C to DM, and H is the foot of perpendicular from M to D C, then the triangles DHM and DN C are similar. Hence MH = C N and DM = D C are equivalent.

6 32 ntonio utierrez, Hiroshi kumura and Hussein Tahir C M N C H D Figure 5. If each of the lines EF, F and E intersects, we denote the point of intersections by U g, U e and U f, respectively for a poristic triangle EF of (,, ). If EF and are not parallel, let δ g = U g (). If EF and are parallel, let δ g be the axis of (,, ). Similarly δ e and δ f are defined. y Proposition 5.1, we get the following theorem. Theorem 5.2. The smallest circles touching and passing through one of the points of intersection of EF and δ g are rchimedean for a poristic triangle EF of (,, ). Similar facts are true for F and δ e and for E and δ f. poristic triangle of (,, ) gives six rchimedean circles in general by Theorem 5.2. Therefore we also get infinitely many sextuplets of rchimedean circles. Figure 6 shows the sextuplet of rchimedean circles in the case in which the poristic triangle is the base triangle. E = I δ e δ g δ f U g U f U e = K J = F Figure Some special cases In this section we consider some special cases in which EF is a special poristic triangle of (,, ). t the beginning, we consider the case the lines and F being parallel (see Figure 7). If a b, let δ = δ e in the case E = I (see Figure 6). The circle δ is expressed by the equation (x+2ab/u) 2 + y 2 = (2ab/u) 2. If a = b, we define δ as the axis. In any case the points ) of intersection of δ and have coordinates (s, ±r (3a + b)(a + 3b)/g. Hence they lie on the Schoch line and also on the circle ( ) [6]. Let

7 Poristic triangles of the arbelos 33 Q be the point of intersection of δ and lying in the region y > 0. coordinates are [5]: ( 2abu (3) w, 2abg ). w Its Let t = w(w + 4ab). The points of intersection of and the line y = 2r have coordinates (4) (u ± t/g, 2r ). E = Q δ E U g F U f F E Figure 7. Theorem 6.1. The following statements are equivalent for a poristic triangle EF of (,, ). (i) The lines F and are parallel. (ii) The point E coincides with the point Q or its reflection in. (iii) F U g = U g holds. (iv) U f = U f holds. (v) The foot of perpendicular from E to coincides with the point ε. Proof. Let us assume (i). We may assume F lies in the region y < 0 and F has coordinates (u t/g, 2r ) by (4). Then F Q is expressed by the equation 2v 2 x + ug3 wt y + gt uw = 0. 2ab The distance between F Q and equals 2r, because we have (gt uw) 2 ((2v 2 ) 2 + ((ug 3 wt)/(2ab)) 2 = 4r2. Hence the line F Q touches. Therefore QF is a poristic triangle of (,, ), i.e., E = Q. Therefore (i) implies (ii). Since E F is one-toone correspondence, the converse holds. The part (i) is equivalent to that the distance between F and is 2r. This happens only when F U g = U g by Proposition 5.1. Hence (i) and (iii) are equivalent. lso (i) and (iv) are equivalent. Let D be one of the farthest points on ε from. The slope of the line Q equals g/u by (3). lso the slope of the line D equals ±r /( r u/g) = g/u. Since E lies on ε and E, (ii) holds if and

8 34 ntonio utierrez, Hiroshi kumura and Hussein Tahir only if E coincides with D or its reflection in. Hence (ii) and (v) are equivalent. If F and are parallel, the point of intersection of δ and one of the external common tangents of and lies on F by Proposition 5.1. Let us assume that E and are the points of intersection of the circles and (), where E lies in the region y > 0 (see Figure 8). Then E and the axis are parallel and their distance equals 2r [2], and the triangle EF is an isosceles triangle, and F coincides with. In this case the points E and are the points of intersection of the circles and [9], where the circle ( ) is denoted by (2) in [9]. The distance between the axis and each of the points of intersection of and ( ) is r [3]. While the points E m and m lies on ( ) by (iii) of Theorem 3.1, and their distance from the axis also equals r. Therefore E m and m are the points of intersections of and ( ). The arbelos (,, ) in Figure 4 is an example of this case. If a = b, then r = a/2. In this case any poristic triangle EF of (,, ) is equilateral and the circles and ( ) coincide (see Figure 9). lso the circle ε touches the rchimedean circles with centers E m at the point E. E F = F m E m E F () E m ε m E ( ) m F F m Figure 8. Figure 9. References [1] Dao T.., Two pairs of rchimedean circles in the arbelos, Forum eom., 14 (2014) [2] Dodge C. W., Schoch T., Woo P. Y., and Yiu P., Those ubiquitous rchimedean circles, Math. Mag., 72 (1999) [3] Lamoen, F. M. v., rchimedean adventures, Forum eom., 6 (2006) [4] Johnson, R.., circle theorem, mer. Math. Monthly, 23 (1916) [5] kumura, H., The inscribed square of the arbelos, lob. J. dv. Res. Class. Mod. eom., 4 (2015) [6] kumura, H., rchimedean circles passing through a special point, Int. J. eom., 4 (2015), [7] kumura, H., rchimedean circles of the collinear arbelos and the skewed arbelos, J. eom. raph., 17 (2013), [8] kumura, H., and Watanabe, M., The twin circles of rchimedes in a skewed arbelos, Forum eom., 4 (2004) [9] kumura, H., and Watanabe, M., The rchimedean circles of Schoch and Woo, Forum eom., 4 (2004)

9 10 LDE RSE CT, THE WDLNDS, TEXS, 77382, US address: Poristic triangles of the arbelos 35 DEPRTMENT F MTHEMTICS, YMT UNIVERSITY, KTYM SUIT SK, , JPN address: okumura.hiroshi@yamato-u.ac.jp 16 LLISN STREET, W. SUNSHINE VIC 3020, USTRLI address: tahirh@oz .com.au