Forum Geometricorum Volume 14 (2014) 397 402 FORUM GEOM ISSN 1534-1178 Congruent Contiguous Excircles Mihály Bencze nd Ovidiu T Pop Abstrct In this pper we present some interesting lines in tringle nd we give some of their properties 1 The e-property For given tringle ABC, nd point X on the side BC, consider the contiguous subtringles ABX nd AXC, with their excircles on the sides BX nd XC We shll sy tht X hs the e-property if these two excircles re congruent (see Figure 1) A B T b X T T c C T c T b T 1 I 1 T 2 I 2 I Figure 1 Denote by, b, c the lengths of the sides BC, CA, AB The rdii of the incircle nd excircle on BC re r Δ s nd r Δ s, where s nd Δ re the semiperimeter nd re of tringle ABC It is known tht BT s c nd T C s b In the following, we shll denote by s T nd Δ(T ) the semiperimeter nd re of tringle T Let I 1, I 2, nd I be the centers of the excircles of tringles ABX, AXC, nd ABC on the sides BX, XC, nd BC respectively, with points of tngency indicted in Figure 1 We denote by ρ the common exrdius of the congruent contiguous excircles of ABX nd ACX Publiction Dte: December 8, 2014 Communicting Editor: Pul Yiu
398 M Bencze nd O T Pop Theorem 1 Let X be the point on BC with the e-property () AX s(s ) Δ (b) ρ ( s+ s ) s ( ) ( ) (c) BX c+ s(s ) ( s+, nd CX b+ s(s ) s ) 2 ( s+ s ) 2 Proof () In tringles ABX nd AXC we hve ρ Δ(ABX) s ABX BX Δ(AXC) s AXC XC Δ(ABX)+Δ(AXC) s ABX BX + s AXC XC Δ AX + s (1) Since I 1 I 2 is prllel to BC, r ρ I 1I 2 r T bx + XT c (s ABX AX)+(s AXC AX) s AX From this, r ρ r s AX, nd AX (s ) ρ r (2) From (1) nd (2), we hve AX 2 (s ) 2 Δ r (s ) Therefore, AX 2 (s ) 2 + (s ) s(s ) This proves () (b) follows from (1) nd () (c) Let BX x Wehve ρ Δ(ABX) s ABX BX Δ(AXC) s AXC XC x c + AX x x b + AX ( x) This reduces to x(b + AX + x) ( x)(c + AX x), nd x (c+ax) b+c+2 AX Mking use of (), we obtin the expression for BX given in (b); similrly for CX Anlogous to the congruent contiguous excircles, the problem of congruent contiguous incircles hs been studied by P Yiu ([3] nd [4]) It ws shown [3, 915] tht for point X on BC, the incircles of tringles ABX nd AXC re congruent if nd only if their excircles on the sides BX nd XC re congruent, ie, X hs the e-property Theorem 2 For the points of tngency of the excircles indicted in Figure 1, we hve () BT b CT c s c s b, (b) AT b AT BX c CX c+ s(s ) b+ s(s )
Congruent contiguous excircles 399 Proof () From the excircle of tringle ABX on the side BX, we hve, mking use of Theorem 1 () nd (c), AX + BX c BT b s ABX c 2 ( ) c+ s(s ) s(s )+ ( s+ c s ) 2 2 (s c) s (3) s + s fter simplifiction Similrly, CT c (s b) s s + s (4) From (3) nd (4), BT b CT c (b) Note tht s c s b This proves () AT b AB +BT b AB +BT b c+ (s c) s s + s ( s c + ) s(s ) s + s Similrly, AT c ( ) s b+ s(s ) s+ s(s ) From these, (b) follows Theorem 3 For the points of tngency of the excircles indicted in Figure 1, we hve () T b T c I 1 I 2 s s+ s, (b) T 1 T 2 b c s s+ s Proof () Since the excircles re congruent, I 1 I 2 T c T b is rectngle Using (3) nd (4), we obtin T b T c BC BT b CT c (s c) s (s b) s s + s s + s ( s + s ) s s + s s s + s
400 M Bencze nd O T Pop (b) If the excircles of tringles ABX nd AXC touch the line AX t T 1 nd T 2 respectively, then, mking use of () bove nd Theorem 1(b), we hve T 1 T2 2 I 1 I2 2 (ρ + ρ ) 2 ( ) 2 s 4Δ 2 ( ) s + s 2 s + s (s ) 2 s 4s(s b)(s c) ( s + s ) 2 s(b c) 2 ( s + s ) 2 Theorem 4 If X hs the e-property, the line AX bisects I 1 I 2 Proof Let the bisectors AI 1 nd AI 2 of ngles BAX nd XAC intersect BC t D nd E respectively In tringle ABX with bisector AD, we hve, by Theorem 1() nd (c), DX BX AX AB + AX s(s ) ( ) 2 s + s Similrly, EX s(s ) ( s+ DX This shows tht X is the midpoint of DE s ) 2 Since I 1 I 2 is prllel to DE, the line AX bisects I 1 I 2 2 Some identities nd inequlities involving points with the e-property Anlogous to the point X on BC with the e-property, there re lso points Y on CA, nd Z on AB with the e-property, ie, the contiguous tringles BCY, BY A hve congruent excircles on CY, YA, nd the pir CAZ, CZB lso with congruent excircles on AZ, ZB (see Figure 2) In generl, the cevins AX, BY, nd CZ re not concurrent If tringle ABC is isosceles, they do, becuse of obvious symmetry Proposition 5 The re of tringle XY Z is ( + s(s c))(b + s(s ))(c + s(s b)) +( + s(s b))(b + s(s c))(c + s(s )) ( s + s ) 2 ( s + s b) 2 ( s + Δ s c) 2 We estblish some identities nd inequlities involving the cevin lines through through the points with the e-property Denote by R the circumrdius of tringle ABC Theorem 6 () AX BY CZ sδ (b) AX 2 + BY 2 + CZ 2 s 2 1 (c) + 1 + 1 4R+r AX 2 BY 2 CZ 2 rs 2 (d) 2(2R r) Δ AX 2 + b BY 2 + c CZ 2
Congruent contiguous excircles 401 A Z Y B X C Figure 2 (e) AX 4 + BY 4 + CZ 4 s 2 (s 2 2r 2 8Rr) (f) AX 6 + BY 6 + CZ 6 s 4 (s 2 12Rr) Proof These follow from Theorem 1(), (c), nd the bsic reltions see [1] b + bc + c s 2 + r 2 +4Rr, 2 + b 2 + c 2 2(s 2 r 2 4Rr), 3 + b 3 + c 3 2s(s 2 3r 2 6Rr); Theorem 7 Let m nd w be the lengths of the medin on BC nd ngle bisector of ngle A IfX is the point on BC with the e-property, then w AX m Proof For the bisector of ngle A, w 2bc b + c cos A 2 2 bc b + c s(s ) 2 bc AX AX, b + c since 2 bc b + c On the other hnd, for the medin on BC, the inequlity AX m is equivlent to s(s ) 2(b2 +c 2 ) 2 4 After some rerrngement, this reduces to (b + c) 2 2(b 2 + c 2 ), which is clerly vlid Theorem 8 () AX BY + BY CZ + CZ AX s 2 (b) 3 3 sδ AX + BY + CZ 3s
402 M Bencze nd O T Pop Proof These follow from pplying the well known inequlities (x + y + z) 2 3(x 2 + y 2 + z 2 ) nd xy + yz + zx x 2 + y 2 + z 2, pplied to x AX, y BY, nd z CZ, nd mking use of Theorem 6 (), (b) Remrks (1) By using Schwrz s Inequlity, we hve ( 1 (AX 2 + BY 2 + CZ 2 ) AX 2 + 1 BY 2 + 1 ) CZ 2 9 By using the formule in Theorem 6(b), (c), we hve s 2 4R+r 9 From this the rs 2 well known Euler s inequlity R 2r follows (2) Agin, it is esy to estblish AX 2 BY 2 + BY 2 CZ 2 + CZ 2 AX 2 (4R + r)rs 2 From the inequlity AX 2 BY 2 + BY 2 CZ 2 + CZ 2 AX 2 AX 4 + BY 4 + CZ 4, nd the formul in Theorem 6(e), we hve (4R + r)rs 2 s 2 (s 2 2r 2 8Rr) This leds to nother known inequlity 3(4R + r)r s 2 ; see [1, 2] References [1] N Minculete, Eglităţi şi ineglităţi geometrice în triunghi, Editur Eurocrptic, Sfântu Gheorghe, 2003 (in Romnin) [2] D S Mitrinović, J E Pečrić, nd V Volenec, Recent Advnces in Geometric Inequlities, Kluwer Acdemic Publishers, Dordrecht, 1989 [3] P Yiu, Notes on Eucliden Geometry, Florid Atlntic University Lecture Notes, 1998 [4] P Yiu, The congruent incircles cevins of tringle, Missouri J Mth Sci, 15 (2003) 21 32 Mihály Bencze: Ady Endre High School, 89 Ferdinnd Boulevrd, Burchrest 021384, Romni E-mil ddress: benczemihly@gmilcom Ovidiu T Pop: Ntionl College Mihi Eminescu, 5 Mihi Eminescu Street, Stu Mre 440014, Romni E-mil ddress: ovidiutiberiu@yhoocom