Trn Quang Hng - High school for gifted students at Science 1. On Casey Inequality. Tran Quang Hung, High school for gifted students at Science

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1 Trn Quang Hng - High school for gifted students at Science 1 n asey nequality Tran Quang Hung, High school for gifted students at Science asey s theorem is one of famous theorem of geometry, we can see it in [3,4]. Ptolemy s theorem (see in [2]) can be considered as special case of asey s theorem but Ptolemy inequality (see in [3]) can be considered as an extension of Ptolemy s theorem. Now we will show an extension of Ptolemy inequality. We begin with asey s theorem. Theorem 1 (asey s theorem). Four circles c 1,c 2,c 3, and c 4 are tangent to a fifth circle or a straight line iff T (12) T (34) ±T (13) T (42) ±T (14) T (23) = 0. where T (ij) is the length of a common tangent to circles i and j. We can see a nice corollary which we call by a part of asey s theorem Theorem 2 (asey s theorem). Let be a triangle inscribed circle (). The circle () touch to () at a point in arc which does not contain. From,, draw the tangents,, to () (,, ()), respectively. Prove that a = b +b. With a,b,c are the sides of triangle, respectively. ' ' Figure 1.

2 Trn Quang Hng - High school for gifted students at Science 2 The folowing theorem is main theorem of this article, it consider as an extension of Ptolemy s inequality. We will call it by asey s inequality Theorem 3 (asey inequality). Let be a triangle inscribed circle (). () is an arbitrary circle. From,, draw the tangents,, to () (,, ()), respectively. Prove that 1/ f () () = then a,b,c are three side of a triangle. 2/ f () () as following () intersects the arc which does not contain then a b +c () intersects the arc which does not contain then b c +a () intersects the arc which does not contain then c a +b Equality holds iff circle () tangents to (). Proof. 1/ f () () =. ssume that radius of () is r, draw circle (,r ) (circle center and radius r ) touch ()at a point in arc which does not contain. Easily seen r r. Draw the tangents,, of (,r ) (,, (,r )), respectively. pply Pythagoras theorem we have 2 +r 2 = 2, 2 +r 2 = 2. Therefore 2 = 2 +r 2 r 2 and analogously then 2 = 2 +r 2 r 2, 2 = 2 +r 2 r 2 (1) ' r r'

3 Trn Quang Hng - High school for gifted students at Science 3 Figure 2. From theorem 2, square both two sides we get a 2 2 = b 2 2 +c bc (2) Now if we prove that b +c a b c then a,b,c will be three sides of a triangle. ndeed, the inequality b +c a are equivalent to b 2 2 +c bc a 2 2 b 2 ( 2 +r 2 r 2 )+c 2 ( 2 +r 2 r 2 )+2bc a 2 2 (Get from (1)) (b 2 +c 2 a 2 )(r 2 r 2 ) 2bc +2bc 0 (Get from (2)) 2bccos(r 2 r 2 ) 2bc +2bc ( 2 +r 2 r 2 )( 2 +r 2 r 2 ) 0 (Get from (1)) cos(r 2 r 2 ) + ( 2 +r 2 r 2 )( 2 +r 2 r 2 ) 0 The last inequality is true ( 2 +r 2 r 2 )( 2 +r 2 r 2 ) +r 2 r 2 because of auchy-schwarz inequality, note that the last inequality is true because cos(r 2 r 2 )+r 2 r 2 0 from r r and (1+cos) 0. We are done. Nowtheinequalitya b c isequivalenttob 2 2 +c 2 2 2bb a 2 2. Use analogous transforms as above we must prove that cos(r 2 r 2 ) ( 2 +r 2 r 2 )( 2 +r 2 r 2 ) 0 ecause ( 2 +r 2 r 2 )( 2 +r 2 r 2 ) (r 2 r 2 ) therefore LHS cos(r 2 r 2 ) r 2 r 2 2 < 0 which is true inequality. The cases (,r ) touch are which does not contain and the arc which does not contain we prove analogously. We are done part 1/. 2/ f () (). ssume (,r) intersect arc which does not contain. Draw (,r ) touch arc which does not contain. Easily seen r r. Draw the tangents,, of (,r ) (,, (,r )), respectively. nalogous, apply Pythagoras theorem as in (1), we get the equalities r 2 = 2 +r 2 r 2, 2 = 2 +r 2 r 2, 2 = 2 +r 2 r 2 2 = 2 +r 2 r 2, 2 = 2 +r 2 r 2, 2 = 2 +r 2 r 2 (3)

4 Trn Quang Hng - High school for gifted students at Science 4 r r' ' Figure 3.

5 Trn Quang Hng - High school for gifted students at Science 5 Use theorem 2 and (3) with analogous transforms the inequality is equivalent to cos(r 2 r 2 ) + 0 (4) Note that = ( 2 +r 2 r 2 )( 2 +r 2 r 2 ) +r 2 r 2 So that LHS cos(r 2 r 2 ) (r 2 r 2 ) = (r 2 r 2 )(1 + cos) 0. which is true because r r,1+cos 0. The cases (,r ) touch arc which does not contain and the arc which does not contain we prove analogously. We are done part 2/. References [1] [2] [3] Roger. Johnson, dvanced Euclidean Geometry Dover Publications (ugust 31, 2007) [4]