An Important Application of the Computation of the Distances between Remarkable Points in the Triangle Geometry
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1 An Imortnt Aliction of the Comuttion of the Distnces between Remrkble Points in the Tringle Geometry Prof. Ion Pătrşcu, The Frții Buzeşti College, Criov, Romni Prof. Florentin Smrndche, University of New Mexico, U.S.A. In this rticle we ll rove through comuttion the Feuerbch s theorem reltive to the tngent to the nine oints circle, the inscribed circle, nd the ex-inscribed circles of given tringle. Let ABC given rndom tringle in which we denote with O the center of the circumscribed circle, with I the center of the inscribed circle, with H the orthocenter, with I the center of the A ex-inscribed circle, with O 9 the center of the nine oints circle, with + b+ c = the semi-erimeter, with R the rdius of the circumscribed circle, with r the rdius of the inscribed circle, nd with r the rdius of the A ex-inscribed circle. Proosition In tringle ABC re true the following reltions: (i R - Rr Euler s reltion (ii OI = R + Rr Feuerbch s reltion (iii OH = r - + 9R + 8Rr (iv IH = 3r - + 4R + 4Rr (v I H = r - + r + 4R + 4Rr Proof (i The ositionl vector of the center I of the inscribed circle of the given tringle ABC is PI = ( PA+ bpb + cpc For ny oint P in the lne of the tringle ABC. We hve ( OA + bob + coc We comute OI OI, nd we obtin: ( OA + b OB + c OC + boa OB + bcob OC + coc OA 4
2 From the cosin s theorem lied in the tringle OBC we get OB OC = R - nd the similr reltions, which substituted in the reltion for OI we find ( R 4 - bc 4 Becuse bc = 4Rs nd s= r it results (i (ii The osition vector of the center I of the A ex-inscribed circle is give by: ( ( PI = - PA+ bpb + cpc - We hve: ( ( OI = - OA + bob + coc - Comuting OI OI we obtin + b + c b bc c OI = R - OAOB + OBOC - OAOC ( - ( - ( - ( - Becuse OB OC = R - nd s= r ( -, executing simle comuttion we obtin the Feuerbch s reltion. (iii In tringle it is true the following reltion OH = OA+ OB + OC This is the Sylvester s reltion. We evlute OH OH nd we obtin: OH = 9R ( +b +c. We ll rove tht in tringle we hve: b + bc + c = + r + 4Rr nd + b + c = -r - 8Rr We obtin s 3 ( ( b( c ( b bc c bc = = Therefore s 4Rs =- + b+ bc+ c- We find tht b + bc + c = + r + 4Rr Becuse
3 it results tht which leds to (iii. ( ( + b + c = + b + c - b + bc + c + b + c = -r - 8Rr (iv In the tringle ABC we hve IH = OH -OI We comute IH, nd we obtin: IH = OH + OI - OH OI OH ( OA + OB + OC ( OA + bob + coc OH ér ( + b + c + ( + b OA OB + ( b + c OB OC + ( c + OC OAù = êë úû b + c + b + c = 3R - -. ( + b+ c b + c IH = 4R -Rr - + b+ c To exress + b + c in function of,r,r we ll use the identity: b + c - 3bc = ( + b + c( + b + c -b -bc - c. nd we obtin b + c = -3r - 6Rr ( Substituting in the exression of (v We hve HI = - HA+ bhb + chc - ( ( HI IH, we ll obtin the reltion (iv We ll comute HI HI = HA + b HB + c HC - bha HB - cha HC + bchb HC 4 - If A is the middle oint of ( BC it is known tht AH = OA, therefore AH = 4R - lso, HA HB = OB + OC OC + OA We obtin: Therefore ( ( ( ( HA HB = 4R - + b + c ( 3
4 It results Similrly, ( + + = b c r Rr HA HB = r - + 4R + 4Rr HB HC = HC HA= r - + 4R + 4Rr HI = é4r ( b c ( b c ( r 4R 4Rr( bc b c ù ê ú 4 ë û ( - Becuse b c ( + - = -, it results ( ( bc -b - c = b + c HI = é b c r 4Rr 4 r 4R 4Rr b c ê ë ( ( ( ( ( ( It is known tht 6s = b + b c + c - -b - c From which we find ( ( ( b + b c + c = b + bc + c - bc + b + c = r + + 4Rr - 4bc Substituting, nd fter severl comuttions we obtin (v. Theorem (K. Feuerbch In given tringle the circle of the nine oints is tngent to the inscribed circle nd to the ex-inscribed circles of the tringle. Proof We ly the medin s theorem in the tringle OIH nd we obtin 4IO = OI + IH - OH 9 ( We substitute OI,IH,OH with the obtined formule in function of r,r, nd fter severl simle comuttions we ll obtin R IO9 = -r This reltion shows tht the circle of the nine oints (which hs the rdius R is tngent to inscribed circle. We ly the medin s theorem for the tringle ( 4I O = OI + I H - OH We substitute OI,IH,OH nd we ll obtin 9 OIH, nd we obtin ù úû 4
5 R IO9 = + r This reltion shows tht the circle of the nine oints nd the A- ex-inscribed circle re tngent in exterior. Note In n rticle ublished in the Gzet Mtemtică, no. 4, from 98, the lte Romnin Professor Lurențiu Pnitool sked for the finding of the strongest inequlity of the tye kr + hr ³ + b + c nd roves tht this inequlity is 8R + 4r ³ + b + c. Tking into considertion tht + b + c IH = 4R + r - nd tht IH ³ 0 we re-find this inequlity nd its geometricl interrettion. References [] Cludiu Condă, Geometrie nlitică în coordonte bricentrice, Editur Rerogrf, Criov, 997. [] Dn Schelrie, Geometri triunghiului, Anul 000, Editur Mtrix Rom, Bucureşti,
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