D. D.DISQUISITIO GEOMETRICA

DE

BISECTIONEANGULORUM in TRIANGULO

RECTILINEO,Quam,

CONSEN. AMFLISS. ORD. PHILOSOPH. UPSAL.,

MODERANTEVIRO CELEB ERR1M0

Mag. FREDERICOMALLET,

Math. Int. Professor. Reg. et Ord. Ac. Reg. Scient.Stockh, et Reg. Sociex Scient. Ups, Membro,

PRO GRADUPUBLICE DEFENDET

OLAUS JOH. FÅHRAEUS,Gothlandus Stipend, Gutiiermuth.

In Auditoric Gusiy Die XII. Maji mdcclxxix.Horis Solitis.

UPSALIM,apud JOHAN. EDMAN 3 direct. et reg, acad. typogr.

§. I.

Figuram Triangulärem elementum omnium plana¬rum & re&ilinearum conflituere, adeo apud Geome-cras in confeffo efle exiftimo, ut fingulorum fcripta

probationis loco cifari poffine. Illa enim omniumnon modo fimplicifllroa habenda, fed ex eadem quo-que reliqu« , utcunque format«, componuntur. Afigura triangulari Pianometria incipitur, & ab ejus pro-prietatibus reliquarum natura eruitur: in arearum di-menfione,quando quadratum five re£tangu'um elementiloco ponuntur, h«c aflumtio, utut arbritraria & vul-gari conceptui aptata, figuris primo parallelogramicis& triangularibus, deinde reliquis applicatur. Trianguliitaque confideratio primum Geometri« Caput ablol-

A 2 vit,

ifp i 4 ( *8*vif,ädeo ut per illam, quafi januam (crentixrellqaa-rum trailationero adire liceat. Ex hoc etiam faäurneft, ut triangulorum ipecialem quandam fcientiaro ad-ornarint Geometri, in qua triangulorum calculus do-cetur, & ejusmodi analyfis inflituitur, ut ex cerris da-tis caetera q jae ufui eflé poterint, deferminentur. Pro-lixior hinc trianguli do&rina enata, & figura haecpliciflima identidem verfa, nova, femper prodit i-eli—quarum principia.

Inter artes vero geometricas, quibus veritates detriangulis elici (olent, illa nobis maxime arrifit, quseper bife&iones angulorum procedit; quaroobrem non-nullas propolitiones, ad eandem maferiam perdnentes,.hic denuo explicare noflrisqui plurimum demonftra-tionibus munire decrevimns; cui innocuo quidem cona-tui, L* H. velie, (ua indulgentia favere enixc rogamus».

§. ir.Theor; L Linea BD (fig. /.), bifecans angnlum

ABC lateri AC oppofitum , determinåi fegmenta e-jusdem latens in proportione laterum angulumABC comprehendentium* h. e. facit AD: DC::AB : BC.

Proprietäten! hancce trianguli dudum nobis detexitEUCLIDES, L. VI. prop. 3.. Geometrarum inlöperoculatiorum fugere non potuhperipicachm3biJeBo'angu-b externo CBE per lineam BF% determinäri AF: CF

V. AB

e§3a ) s> f «S&

r: AB r BG; Dufta enim CG parallela cum BF eritangulus BGC-rz EBF, & BCG atqualis CZ?F; adeoqueBGC atqualis BCG\ imo BG zzz BC\ Huic erit äF\FC : : AB : BG : ; AB : BC :; AD : DC. Ex eadenavero condru&ione iequentia accipias corollaria,CorolL f . BF femper erit nocmalis ad BD..Coroll 2. Si fuerit BF parallela cum AC, erit triangu-

lum ABC sequicrurum, five AB = BC, & ob ean-dem rationem AD = DC*

CovoU. 3. Si BF cadat ad partes punfti C, erit AB> BC»& vice verfa, pofito AB> BC, (ecabit BF linearaAC produåam ultra C. Contra vero erit AB<BQ,få BF lecuerit AC ad partem pun&i A & conyerfim.

§. rir..Theo?. 2. S/ angulus ABC (Fig. 2.) bifecius

fuerit a linea BD, erit quadrafum line<ß angülumbijecantis, äquale excefui rettanguli a lateribusJupra illud, g&od 0 bajeos fegmentis continetnr,five BD2 = AB .BC — AD .DC.

Circa triangnlum defcribatur circulus AKBCHr,& produeatur B-D ad peripheriam Circuli in //", jun-gaturque //C. Jam vero ABD zzzHBC» & BÅDz-z BHC»adeoque triangulum «quiangulum cum illoBHC erit itaque AB: BD : : BH:: BD;, live ^

A. 3; =

cSIJ \ f\ (J u \

j=s BD ,BH. Sed BH.BD =2 BD.DH 4- ^Z)2, &FZF FZ/ = yiF . DC; Ergo F^ . CB z=i AD. DC -ArBD\ five AB. BC — . FC = BD2 i

Similiter: BifeSlo angulo externo CBEper lineam BFamV quadratum Imex unguium extermim CBE bifecantis3xquäle exceffui reSlangali a Jegmeutis bafeos , fupra illui,quod a latertbus continetur, five BF2 = ./ZF . FC —•

^F, BC.

Producatur enimFFad occurfum cum peripheria inZT3dein jungatur erifque FCF live angulus exteriörquadrilateri KACB aequaiis angulo interiori & oppofi-to ARB, hinc elucec triangulum aKB efie sequian-gulum cum triangulo BCF adeoque AF:FK:: BF: BClive AF, FC m BK . BF, Ted FK. BB =1 BF2 +BF .BK; & ob ABKzxz EBB = CBF erit triangulumABK requiangulum cum BCF, adeoque AB: BK ;:BF: BC; hinc AB.BCzzzBK. BF, & FF2 FF. FZZ=BF2 + ^F, FC= 4F. FC five BF2 z=z AF .FC —

AB , BC,

CoroU. 1. Quoniam DBF eft requalis angulo reäo, eritangulus KBH etiam rcwlus, adeoque KH\ diameterCirculi AHCBt

CoroU. 2. Linea KH erit normalis ad AC> nam AH-=.HC& HK diameter, ergo bilecabit .chordam ACi eidem-que normaliter infiftet.

CoroU.

ty C effcy 7 v. w

Coroll 3. Si latus AC befeäum fuerit in G, eidemquenormalis fiat linea KGH> occurrens in //, & K lineis BD, BP, angulum internum ABC & exter-nutn CBE bifecantibus, erunt pundta H, Ky in cir-culo triangulum ABC circumfcribente.

SchoL Ex a Ila tis nemini non patere arbitror, intcr an¬

gulum trianguli dati internum, & externum, mutuamreperiri dependentiam atque relationem, eandemquein proprietatibus trianguli, quas, ab angulorum bife&io-ne ulterius derivantur, inveftigandis minus omitten-dam cfTe. Quemadmodum enim ex cujusvis anguli bi-fe&ioney theoremata addudta ufui inlerviunt,ita'duosangulos bifecando, ambo videlicet vel infernos velexternes yVel denique unum internum alterum ex¬ternum modo diveriumvaria, attentione utiquedigna occurrunt, quaf vel naturam trianguli diluci-dabunt, vel in ejus refolutione multum adjumentiiadferento-

§. IV.Theor. /. In friangulo ABC f Fig. 3.) ß Anguli

interm BAC, ABC bifeccntur a hneis AD, BD invi-cem occurrentibus in punäo D, linea CD jungemtertium angulum ACB cum pundo concurfus Dybifecabit etiam eundem angulum ACB.

A punfto enim D demittantur ad latera AB, AC} CBnormales DEy DFXDG; & quoniam triangula ADE,

ABF

n ) s r &JDF fune reaangula in E, F, habentque angulosEAD, DAE äquales, una cum latere communi AD,erunt eadem triangu'a Tibi mvicem perfekte ^qualia, &hinc DE zrz DFy AB zz AF. Eadem ratione demon«flratur eile DE=DG, & BE=z EG. Quoniam ergoDF=z DE = DG & DC communis in triangulis CFDtDGC,etit etiam CG~CF, & angulus GC'D-zr. ACD.

CoroU. i. Si itaque centro D, atque radio £)£, deferi-batur circuius, erit hic triangulo ABC inferiptus.Cfr. Eucl. L* IV. prp. 4.

CVo#. 2. Bife&is duobus angulis incernis BAC,per lineas ^Z), Z?Z>, & a pundto eoneurfus earum Ddemifla normaii DG vel DFy erit fegmentum GCVel FC inter normalem & tertium angulum ACB,aequale exceflui femilummse laterum trianguli fuperlatus yZ#,jacens inter angulos bile&os. NamAE-=AF,BE = BG, CF=z CG; dat AE + BE = AB = AF-4- BG adeoque AB «V Z?(7 ■+ CG. Di-cantur AB -+• BC -f- ZfC = 2 «9, erit -f CY7 = Sadeoque CG =r CF=5 — AB. Eadem ratione AF+ BG + GC = S, five AF—S—BC, & i?£=== S—AC

Corol. 3. Si duo anguli externt (Fig. 4,) ut FAB, ABGbifecentur lineis AD, Z?Z) ab enrum concurfu Dducatur DC, eni angulus ACB iterum bifeSlus. Ne»que minori binc conficitur evidentia, y? angulus ex-

9 ) 9 ( Ofernus FÅB internus ACB linets AD, C73 jfo Dconcurrentibus bifecentur jun&a yidelicet DB, bifecariquoque angulum externutn ABG. Eft namque demon«ftrandi methodus eadem, quam in ipfa propofitio-ne, demiffis normalibus DF, DG, DE, dedimus.

Coroll. ^.Data corollarii praecedentis conftruäione eritCF vel CG gequalis femifummae laterum hujus tri-atiguli ACB, five CF~ C(7= S; nam CFzn. CA -\-AE,CG=zC3 + BE:

Coroll. 5. Sumto D pro centro, circulus per E du&us,tanget latera -trianguli ABC in punåis Ft E, G.

Coroll. 6. Segmentum ab angulo externo bife&o adperpendiculum DF, ut AF asquale eft exceflui femi-fummae laterum tuper latus AC five CF^sz S— AQ.& 3G — S—BC

Coroll. 7. iisdem pofitis erit S—ABzziAC—(72?=5BC—AF, five GBzzABAr AC—S9 AFzz A3±BC—S.

§. V.Tbeor* 4. In triangulo ABC (Fig. <.) fi fuerii

€irculus EFG infcriptus, & a centro circuli D du«tantur linea DA, DB, DC, erunt anguli, A,B,C,ab iisdem bijeftu

Ducantur sd pun£ia conta&us E,F,Gy radii DE,DF, DG; in triangulis itaque ADE, AFD reaangu-

3 ii*

SI ) 10 ( uDs ad E, Fy latus AD commune habetur, & lateraDE, DF aequalia deprehenduntur, ergo AE-=z/lF\ at-que angulus DAEzxzDAF. P*ari ratione oftenditur eileDCFz=z DCG, atque DBE = DBG.

Si circulus EFG contingat låtera tviangult Abc extraipfum , linets DA, Db, Dc bifecabunt angnlos hAc inter-mm y cbßr bcC vero externas. Du&a enim Dg ad pun¬ctum concaftas g, demonftratio mediantibus lineis-fl^DF antea du&is, eadem ratione concinnatur..

f. vr.Tbeor. y. Si line# AD, BD, CD C Fig. 6.) Infe-

vantes angnlos internos trianguli ABC cmcurrantin Dy & fiant BE, CF normales ad produäamAD, atque DG ad BC crit produäum normaliumBE, CF äquale reäangulo fub fegmentis laterisBC, ßve BE . CF = BG .GC. Et produäum feg-mentorum DF, DE äquale quadrato normalis DG,radii fcilicet in circulo, qui triangulo ABC in-fcribitur, k. e. DE. DF = DG2i

Eft enim angulus FCB ~ CBE, & ACF = ABFfive ACB — FCB = ABC -+- CBE, ergo ACB = ABCHh 2CBE; unde etiam' £ ACB == ~ ABC CBE, fiveDCGzz DBE\ adeoqtie rriangula DCG, DBE aequian-gula , Se BE : BD :: CG : DC, nec non DE : BD: :DG : DC; fimiliter erit ■§■ ABCz=z ± ACB — CBE fiveDBGz= DCF, unde BD BG : : DC: CF, & BD: DG

DC

& j " r #?: DC: DFs hioc BE: BG :: CC:CF; atque DF: TOt: T)C.- T)F, adeoque FF * CF zz BG . CG; & DE,DFzz DG* 5

Coroll. i. Quoniam CG zz S^ÅB^Sc BG zz S—• y/C5/w3§. 7 K 77><?or. 3 , Cor*/. 2, erit BE. CF zz(S~AB)(TfC).

Coro//. Hinc AC.AB: (S—^F) (S—AC) ::( 7iW)a : ( Sin 4- BAQ2; nam ^F: FF : : &*</.-5V« 4. F^C ,& AC: CF:: Rad : Ä» | F4C, adeoqueAF-.ilC : BE.CF : iAB MC\CG . GB: : (Rad)* :(ä»4- BAC)*, quo mediante theoremate, dimidiusarsgulus Fi4C, ex datis trianguli ABC lateribus, fup-putari poteflv

Coroll. 3. Si linea DA, 727?, 7)C, (Fig. 7.) bifecantesangulos extevnos CBH, BCK, £f internum BAC con*currafit in T), ducantur normales DGy BE\CF\utin pYopofitiove monuimus, erit CF.BEzzBG . GC atqueDE. DF zz TO* ; nam FCK zz iDCG 4- FCF=

EBH zzz GBD -CBE-, ergo DcGzzGBD— CBEfiVe DCGzz FFT), & GBD zz DCG + BCFzzDCF;unde etiam triangula 7)CG, FFT) sequiangula erunt,fimulque triangula GBDyDCFfimilia'hine, ut ex allatispatet, erunt FF: TiC:; GC: CF, & T)F: T)C: .* T)C;T)F; five FF. CFzzBG.GC, & DE. DFzz DG2\

Coroll. 4. Demifils normalibus T)/7, D/f, zftBGzzBH h,CGzzCK, ergo etiam CF. FF zzCK. BH. Prseterea

B » iX?

® ) Ii ( iDG = DH = AF adeoque AF.. AF := DR,DKzzDH*-,

CovqU, 5. Si FA, BD (Fig. g.) bifecantes angulosA^CABC concurrant in A, & praeter normales i?#, CFin AD y fiant normales CF, AR, ad lineam Z?A,atque DG, AL ad latera BC erit AF, AF =DGa = AZ2 .= AF, AA;

§. VII.Theor. 6. Z» triangulo ABC, ( Fig. 9.) fi angulm

BAC bifeftus fuerit a linea AD, eidemque norma li¬ter inßßant linea BE, CF, erit produRmn jegmen-torum AF, AE aquale reffangulo fub fimifummalaterum S& eadem Jemifumma & låter e BC,quodangulo BAC opponitur, dwiinuta, h. e. AF. AE =S( S. — BC).

Fiat €BD = DBH, & FLzzzLE, prodücatur CFad occutfum K cum AB, & fint DH, DG, LM nor¬males ad AR, BC, AD, dein jungantur FM, ME.Jam itaque AB + BH •= AB -{- BG = AH .= S(Cor. 4. Theor. 3, §. IV.) & AB BCzu S -f- CG, flveAB — CGzz S — BC; KF =FC & KM = MB, er¬go FM=z ^BCs & quoniam AK .= FC = 5 — CG,erit S — AC= S— AK—AH—AK = CG z= KR; atqueKR + BG = FA + FA= BCzz 2 MH, five MH=z±BC zz FM; verum FMzz ME, fi ergo centro M de-Icribatur circulus per H, idem tranfibit per Ff at¬

que

) 13 <

qoe F,ex quo tandem conficitur AF .AE — AH;MFZTMH= AH . AB — CG=

CoroU. x. Quomara AC: AF: AB : AE, adeoqueAC. AB :AF. AEl: AB2.• AE2:;{Rad)* :{Cof±BACf,erit AC . AB : S . S — BC .*■ * ( Bad )a • Cof%(_ BAC)*;Unde daiis in trjangulo ABC oronibus la-tecibus, poterit Cofinus dimidii cujusvis anguli cal-cuio deterrainari*

Coroll. 2. Eft quoque praeterea 5— AB. S — AC:AC. AB:: (Sin * BAC)» .♦ (Rad)* (Coroll. 2, Tbeor. f.

§. VI.) & Co/; 5» .• .• Rad Tag; Ergo S. S — BC:S— AB .S — AC:: (Rad)*: (lang* BAC)2 : t(Cot. * BAC)2 (Rad )2,- ex quo omnibus darum ,datis omnibus trianguli lateribus tangentem velcotangentem dimidii cujusvis anguli inveniri pofle«,

§. VIII.Schol. Quanquam in Theorematibus quarfo & quui«

to Titus linearunx admodum variari poftunt, proutanguli interni vel exierni biiedi fuerint» relationes ta¬rnen fegmentorum per eadem facile determinari quis.non videt? Tic dato triangulo ABC(Fig. 10.) &EBDbifecante angulum externum CRHy du£tis normalibus;AEy CD ad lineana EBD f & BG cum AE vel CDparallela erit angulus, internus ABC per reftam BG bi-Teaus & BD . BE = CF. AG = S—AB. S —BC*(Coroll. u Theor. j. §. VI).

B.% S2-

^ ) 14 c #Simul quoque erit AB. CD =BG . BF= S. S.—AC

( Theor. 6. §J,VII.)§. IX..

Theor. 7. Pofitis iisdem conßru&ionilus ac in §. 1FTerit in Fig. 3. AD* — £C2 = AC ( — FC) ,• fi-triiliter in Fig. 4. FD2 —^Z>a z=z.AB..AC — BCrnee. non CDX —AD2 = .dC. ABBC..

Nam Fig. 31a j4Z)2—Z>Ca =: ^4Fa' —'.FC* =3

FC, — FC; fed ,4F + FC = AC\ AF =£ — FC,; FC = 5 —^C, adeoque AF—FC =z AC— BCj ergo ^Z)2 Z)Ca =2 AC. AQ — ßC.' InFig. 4. eadem demonftrandi methodo offenditur efle

BD2 — AD2 = BE2—AE2 = AC\ Jß~BC.. Si¬mul vero patet efle CD1 — AD2 = CF2—AF\ &ob CF—- AF =: AC} CF 23 S, AFzzz S-— ACy adeo¬que CF-4- /1F= 2«?—AC^z BC -+- /IF habetur CD%

— AD2 =2:AC..AB + BC,.TANTUM..