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Triangle and Its Circles from the Standpol T of Modern A TRIANGLE A*D ITS CIRCLES nCLU MAT COOK Hlrtn ., College, 1913» B.8., Phillips Uhivoraity, 1925 i wmaa submitted In partial fuirillwmt ©f the requlreBenta for the of MatKB OF SCIXSCB XAR8A8 STAK AORICaLTORAL COLLEGE 1920 —— — ——- DoccP rncn+ C65 ttMM OF UWIUM 3 I fetation 4 ELK* BR1ST SBVZSW 0FCIRCLB8 WOM 20 TO A TMAWLB in mrntscr asoBuwr ——— 6 THK TrUASBLB AHD ITS CIRCLES FROM THK STAKDPOT?IT OF QDKM 0K1KETHY 7 Properties of • Triangle used In Connection with Circles Pedsl or Siraeon Line e n s — — 8 8 i —— — . ntlperallela — SjRaaedlon Line end Synwedien Point —— 10 Isogoml Conjugates ~ 12 Appolonius' Theoren lb Circles 18 Pe<:el Circle » — —— lusscribcd and Inscribed -—————-- 21 Zaerlbeu sad Inscribed — — — 28 Hine-polnt Circle » »— 30 "~ 34 P Circle — Feuerbach's Theorem » ' -~ 37 T-emoine's First Circle or Triplicate Satlo 40 Leaoine's Leeond Circle or Cosine Circle ~ Broeard Circle i Circle cr Circle - ?olar Circle «» •s Circle Other Properties of the irlangle lev Um — 71 telner's Point — 74 Terry's Point — it Inversion of a Triangle (a) With Bespeet to Inscribed Circle — 76 ith nespeot to Clrcwsseribod Clrele — 7« (e) If eapeot to Kserlbed Circle -—»-— 77 ACKWOWLKDOFEHT 81 MM Mi 82 ItTRODUCTIOl This dissertation Is an attempt to present a consistent and systematic account of son* of the various theories of the triangle and Its circles from the standpol t of modern geometry end to solve a number of original exercises in connection with these theorem*. In the science of geometry the triangle and circle have always played an 1 portent role. However, a wonderful nd» vonce has been made in this subject during the latter half of the Nineteenth Century. That new theorems should hare been found In recent time la remarkable when we consider that theoe figures were subjected to close examination by the areeka and the many geometers who followed then. Recently developed properties have been added by Stelner, Fenerberg, Lesoine, Seuberg, Crelle, Jacobl, Hrocerd and a countless mmber of others. Crelle (1780.1865) a Oerman mathematician who first dis- covered the Brocard points said, "It is i~deed wonderful that so si -pie a figure as the triangle Is so inexhaustible in prope ties." This treatise is intended to give Interesting facte concerning the different circles associated with a triangle. A complete account would be i possible but some of the recent- ly developed properties of the triangle and circle will be BwaaaaawasBSBi One of the outstanding features of modern geometry as contrasted with ancient or Euclidean ia that on the whole It is descriptive rather than metrical. In elementary geometry the triangle ia considered as a portion of a plane bounded by three straight lines. In modern geometry the triangle is considered as a system of three non-eoncurrent Infinite lines in which the three intersections are taken Into account or as a system of three points which are not collimear in which the three atraight lines of connection are taken into account. notation In general the following notation will be used for the triangle ABC. ,B,C, -vertices of triangle. A'.B'.C 1 ,- -aid-points of sides. ,P, feet of the altitudes. 0- Centroid. —- ~ —-— —orthocentre -circtt centre 0'- — ——--incentre I1»^2» I3 exeentrea X,Y,Z —points of contact of the in- circle. X«,T',^' —points of contact of the ex- circles on BC.CA.AS respect- ively. r radius of inscribed circle. R- —-—radius of circumscribed circle. r l> r2» r3 ex- radii. a,b,c, —-lengths of sides. a- -_--- —semi perimeter. X —HBagel point. K— -—syjsmedian point. &R- —— radius of nine-point circle. BEIEF REVIEW OP CIRCLES RELATED TO A TRIAHOLE IH ELEMEBTARY GEOMETRY AKD TRIOOBOi'.ETRT In eleaentary geometry are found many facts concerning the inscribed, circumscribed and escribed circles. To find the radii of theee circles In terns of the sides of the triangle furnishes an interesting study. For r « f(s- a)(s - b)(a-c) ' 8 R * abc ^ats-a) (s-b) (a-c) r = l J8(s-a) (s-o) (s-c) s-a - r2 ^(a-aHa-bHa^T) s-b r " 3 ^O^aMs-b) (a-c) 8-C Y Then many theorems are baaed on the relationship between these circles and the parts of the triangles concern- ed. A useful theorem in plane geometry states that in any triangle the product of any two sides is equal to the dia- meter of the circumscribed circle multiplied by the perpen- dicular drawn to the third side from the vertex of the opposite angle, (ab * a Rh) In trigonometry the sine law may be developed froi the circumscribed circle and the tangent law from the inscribed circle. as 2 R S b = o a aXai sin sin and tanjjok= r s -a tan^fis r T=b- tani^f" r s -c THE TRIAHOLE AM) ITS CIRCUS FROii THi. STJWDPOIKI 01 HODiBS 8B0METHY PiC. Fig.2. Pedal or Slmaon Line If from any point P on the clrcuricircle, perpendiculara PL, PM, PH are drawn to the sides of a triangle, then LrJH la a straight line and la called the Pedal or Slrason line. In Pig. 1, page 7. From P any point on the circumclrcle perpendiculara PL, PH, PS are drawn to BC, CA, AB res active- ly- To Prove* LHH Is a straight line. In the quadrilateral BLHP, BHP and BLP are right angles. Than B, L, N, P are concycllc points and LNP and LBP are supplementary anglea. / lso since PHAH are concycllc polnta angle PM - PAH (measured by one-half arc PH) angle CAP * LHP (both supplementary to LBP) How angle CAP and PAH are sup; lementary Then angle LRP and PHH are supplementary Therefore LNH Is a straight line. Antlparallela In the triangle ABC (Figure 2, page 7) the line JL la drawn making the angle LJA S ABC. Then angle ALJ equals ACB. And LJ Is antlparallel to BC. This term "antiparallel" la due to i^elbniz. 3) c. £, >M An Interesting fact to note is that h, J, C. B, are coneyclic points aa the opposite nnglea In the quadrilateral BCJL are supplementary. Syaiaedian Lines and Syamedian Point A line drawn through the vertex of a triangle so as to bisect the antlparallels to the base is called a synmedlan line. The point of concurrence of the three symnedians Is called the synswdian point. In 1873 Eraile Lemoine called attention to this point which since has been referred to as point de Lemoine in French works; nrcbo's point in Genua works. In this pr.per the English term synraedian point will be used and will be indicated by K. Its distances from the sides of the triangle are proportional to the lengths of the corresponding sides. In Pig. 3, page 9. If PQR be the triangle formed by tangents to the circum- circle at A, B, C then AP, B"), CR, bisect all the anti- parallels to BC, CA, AB respectively and are concurrent. First: To Prove: AP, BQ, CR are concurrent. CP "C AO RA BR multiplying (PB) (OC) (RA) - (BR)(AQ) (CP) Therefore AP, BQ, CR are concurrent (Cave's Theorem) This point of concurrence is the sywaedian point. Second: To Prove: AP, BQ, CR bisect all antiparallels to BC, CA, AB. Draw the antlparallel MP to BC through p S PH Is parallel to "AP. (angle PMA * BAQ » CBA) Angle PSB = KAE RBA = SBP Therefore PC PB Similarly PH PC How PC PB Therefore SP • TU. and P Is mid point of US and all antl- parallels to DC would bo bisected by BO; all antiparallela to BA would be bisected by CR. At, BQ, CR ere called symnedian lines. It is important to note that all antiparallols to 3C are parallel to tangent at A; all antlparallels to CA are parallel to tangent at Ej all antiparallols to AB are parallel to tangent at C. Fig. 4. Isogonal Conjugates If from two points lines are drawn to the sane vertex of a triangle and these lines make equal angles with the adjacent sides of the triangle these two points are called Isogonal conjugates* Fig. 4, page 12. In the triangle ABC, the cent ro Id, and K, the symmedlan point are found. To prove K and are isogonal conjugates. If it can be proven that angle OAB • KAC than K and will be Isogonal conjugates. Draw any antlparallel US to BC. Fig. 5. AX bisects MS In B. Sines triangle* ASK and ABC are similar AB> BC » AM: US also AB: BA» AM: Ma. Then the triangle ABA' la similar to AM* Therefore angle BAA' ~ MAm or angle OAB KAC making K and isogonal conjugates. ..xerciae I . Tne isogonal conjugate with regard to an angle of a triangle of a line through the vertex of that angle is perpendicular to the Simeon line of the second intersection of the given line with the circumscribed circle* In Pig. 5, page 13. Given the triangle ABC in clrcunscribed circle. Draw a line AS and find almson line of S which is IJfl. Then draw AO the isogonal conjugate of AS with regard to angle A. Then angle BAQ s SAL To prove: AQ is perpendicular to Simeon line Ut. Angle SBC SAC (measured by \ arc CS) B, B, M, | are eoncyclic points. Therefore angle SHM SBM But angle BAQ • SAC ( isogonal conjugates with respect to angle A. Then angle SBC BAQ and angle SHM = BAQ Then sine* SUA is a right angle and angle SUM * BAQ Therefore HL is perpendicular to AQ. 15 Pig. 6. 16 2 2 2 2 To Prove: AS 135 « 2 IE * 2 BT when L is aid point of BC.
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