On Triangles with Vertices on the Angle Bisectors

Transcription

1 Forum Geometricorum Volume 6 (2006) FORUM GEOM SSN On Triangles with Vertices on the ngle isectors Eric Danneels bstract. We study interesting properties of triangles whose vertices are on the three angle bisectors of a given triangle. We show that such a triangle is perspective with the medial triangle if and only if it is perspective with the intouch triangle. We present several interesting examples with new triangle centers. 1. ntroduction Let be a given triangle with incenter. y an -triangle we mean a triangle UVW whose vertices U, V, W are on the angle bisectors,, respectively. Such triangles are clearly perspective with at the incenter. W P V U Q Figure 1. Theorem 1. n -triangle is perspective with the medial triangle if and only if it is perspective with the intouch triangle. Proof. The homogeneous barycentric coordinates of the vertices of an -triangle can be taken as U =(u : b : c), V =(a : v : c), W =(a : b : w) (1) for some u, v, w. n each case, the condition for perspectivity is F (u, v, w) :=(b c)vw+(c a)wu+(a b)uv +(a b)(b c)(c a) =0. (2) Publication Date: October 16, ommunicating Editor: Paul Yiu. The author thanks Paul Yiu for his help in the preparation of this paper.

2 248 E. Danneels Let D, E, F be the midpoints of the sides,, of triangle. f P =(x : y : z) is the perspector of an -triangle UVW with the medial triangle, then U is the intersection of the line DP with the bisector. t has coordinates ((b c)x : b(y z) :c(y z)). Similarly, the coordinates of V and W can be determined. The triangle UVW is perspective with the intouch triangle at ( x(y + z x) Q = : s a y(z + x y) s b : ) z(x + y z). s c onversely, if an -triangle is perspective with the intouch triangle at Q =(x : y : z), then it is perspective with the medial triangle at P =((s a)x((s b)y +(s c)z (s a)x) : : ). Theorem 2. Let UVW be an -triangle perspective with the medial and the intouch triangles. f U 1, U 1, W 1 are the inversive images of U, V, W in the incircle, then U 1 V 1 W 1 is also an -triangle perspective with the medial and intouch triangles. Proof. f the coordinates of U, V, W are as given in (1), then U 1 =(u 1 : b : c), V 1 =(a : v 1 : c), W 1 =(a : b : w 1 ), where u 1 = (a(b + c) (b c)2 )u 2(s a)(b c) 2 2(s a)u a(b + c)+(b c) 2, From these, v 1 = (b(c + a) (c a)2 )v 2(s b)(c a) 2 2(s b)v b(c + a)+(c a) 2, w 1 = (c(a + b) (a b)2 )w 2(s c)(a b) 2 2(s c)w c(a + b)+(a b) 2. 64abc(s a)(s b)(s c) F (u 1,v 1,w 1 )= cyclic (2(s a)u a(b + c)+(b F (u, v, w) =0. c)2 ) t follows from (2) that U 1 V 1 W 1 is perspective to both the medial and the intouch triangles. f an -triangle UVW is perspective with the medial triangle at (x : y : z), then U 1 V 1 W 1 is perspective with ((y + z x)((a(b + c) (b c) 2 )x (b + c a)(b c)(y z)) : : ), (a((a(b + c) (b c) 2 )x (b + c a)(b c)(y z)) : : ).

3 On triangles with vertices on the angle bisectors 249 Theorem 3. Let UVW be an -triangle perspective with the medial and the intouch triangles. f U 2 (respectively V 2 and W 2 ) is the inversive image of U (respectively V and W )inthe- (respectively - and -) excircle, then U 2 V 2 W 2 is also an -triangle perspective with the medial and intouch triangles. f an -triangle UVW is perspective with the medial triangle at (x : y : z), then U 2 V 2 W 2 is perspective with ( (s a)(y + z x)((a(b + c)+(b c) 2 )x +2s(b c)(y z)) : : ), ( ) a s a ((a(b + c)+(b c)2 )x +2s(b c)(y z)) : :. 2. Some interesting examples We present some interesting examples of -triangles perspective with both the medial and intouch triangles. The perspectors in these examples are new triangle centers not in the current edition of [1] Let X a, X b, X c be the inversive images of the excenters ( a, b,) c in the incircle. We have X a = r2 a and a = a = s s a 1 = a s a. Hence, X a = a 2 r 2 (s a) = a sin 2 ( 2 ) = and by symmetry X a = X b = X c X a X b X c are homothetic with ratio 4R : r. abc (s a)(s b)(s c) = 4R r, = 4R r. Therefore, triangles and X a =(a 2 + b 2 + c 2 2ab 2bc 2ca)(a, b, c) +(b + c a)(c + a b)(a + b c)(1, 0, 0), X b =(a 2 + b 2 + c 2 2ab 2bc 2ca)(a, b, c) +(b + c a)(c + a b)(a + b c)(0, 1, 0), X c =(a 2 + b 2 + c 2 2ab 2bc 2ca)(a, b, c) +(b + c a)(c + a b)(a + b c)(0, 0, 1). Proposition 4. X a X b X c is an -triangle perspective with P x =(a 2 (b + c)+(b + c 2a)(b c) 2 : : ), Q x =(a(b + c a)(a 2 (b + c)+(b + c 2a)(b c) 2 ): : ).

4 250 E. Danneels c X b b X c P x Q x X a a a Figure Let Y a, Y b, Y c be the inversive images of the incenter with respect to the -, -, -excircles. Y a =(a 2 + b 2 + c 2 2bc +2ca +2ab)( a, b, c) +(a + b + c)(c + a b)(a + b c)(1, 0, 0), Y b =(a 2 + b 2 + c 2 +2bc 2ca +2ab)(a, b, c) +(a + b + c)(a + b c)(b + c a)(0, 1, 0), Y c =(a 2 + b 2 + c 2 +2bc +2ca 2ab)(a, b, c) +(a + b + c)(b + c a)(c + a b)(1, 0, 0). Proposition 5. Y a Y b Y c is an -triangle perspective with P y =((b + c a) 2 (a 2 (b + c)+(2a + b + c)(b c) 2 ): : ), ( a(a 2 (b + c)+(2a + b + c)(b c) 2 ) Q y = b + c a ) : :.

5 On triangles with vertices on the angle bisectors 251 b b c c Y b c Y c Q y P y b c a b Y a a a Figure 3. a 2.3. Let V a, V b, V c be the inversive images of X a, X b, X c with respect to the -, -, -excircles. V a =(3a 2 +(b c) 2 )( a, b, c)+2a(c + a b)(a + b c)(1, 0, 0), V b =(3b 2 +(c a) 2 )(a, b, c)+2b(a + b c)(b + c a)(0, 1, 0), V c =(3c 2 +(a b) 2 )(a, b, c)+2c(b + c a)(c + a b)(0, 0, 1). Proposition 6. V a V b V c is an -triangle perspective with P v =(a(b + c a) 3 (a 2 +3(b c) 2 ): : ), ( a(a 2 +3(b c) 2 ) Q v = b + c a ) : : Let W a, W b, W c be the inversive images of Y a, Y b, Y c with respect to the incircle.

6 252 E. Danneels b b c c V b c V c Q v P v b c a b V a a a a Figure 4. W a =(3a 2 +(b c) 2 )(a, b, c) 2a(c + a b)(a + b c)(1, 0, 0), W b =(3b 2 +(c a) 2 )(a, b, c) 2b(a + b c)(b + c a)(0, 1, 0), W c =(3c 2 +(a b) 2 )(a, b, c) 2c(b + c a)(c + a b)(0, 0, 1). Proposition 7. W a W b W c is an -triangle perspective with P w =(a(a 2 +3(b c) 2 ): : ), Q w =(a(b + c a) 2 (a 2 +3(b c) 2 ): : ).

7 On triangles with vertices on the angle bisectors 253 c W c P w W b Q w b W a a a Figure 5. Reference [1]. Kimberling, Encyclopedia of Triangle enters, available at Eric Danneels: Hubert d Ydewallestraat 26, 8730 eernem, elgium address: eric.danneels@telenet.be