ATRIANGLE A*D ITS CIRCLES
nCLU MAT COOK
., Hlrtn College, 1913» B.8., Phillips Uhivoraity, 1925
iwmaa
submitted In partial fuirillwmt ©f the requlreBenta
for the of
MatKB OF SCIXSCB
XAR8A8 STAK AORICaLTORAL COLLEGE
1920
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Touches the sides of the triangle, whose vertices are half-way from the Hagel point to the given vertices, at the points where each meets.
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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
DoccP
rncn+
C65 (^) ttMM OF UWIUM
I fetation BR1ST SBVZSW^ 0FCIRCLB8^ WOM
mrntscr asoBuwr^ ———
THK QDKM TrUASBLB 0K1KETHY^ AHD^ ITS^ CIRCLES
20 TO A^ TMAWLB^ in^ ELK*
FROM THK STAKDPOT?IT OF
Properties of •^ Triangle^ used^ In^ Connection^ with Circles
Pedsl or Siraeon Line e n s —^ —
. ntlperallela^ i^ ———^ —
SjRaaedlon Line^ end^ Synwedien^ Point^ ——
Isogoml Conjugates^ ~
Appolonius' Theoren Circles
Pe<:el Circle »^ —
lusscribcd and Inscribed^ -—————--
Zaerlbeu sad Inscribed —^ —^ —
Hine-polnt Circle » »—
P Feuerbach's Circle^ — Theorem^ » ' "~-~
T-emoine's First Circle or Triplicate^ Satlo
Leaoine's Leeond^ Circle^ or^ Cosine^ Circle^ ~
Broeard Circle cr^ i^ CircleCircle -
?olar Circle •s Circle^ «»
Other Properties of the irlangle
lev Um —
telner's Point —
Terry's Point —
Inversion of a^ Triangle
(a) With Bespeet to^ Inscribed^ Circle^ —
ith nespeot^ to^ Clrcwsseribod^ Clrele^ —
(e) If eapeot to Kserlbed^ Circle^ -^ —»-—
ACKWOWLKDOFEHT
lb 18 21 28 30 34 37 40
MM Mi
it 76 7« 77 81 82
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 I (^) 1»^2» I 3 exeentrea
X,Y,Z —points of contact of the in-
X«,T',^' circle.
—points of contact of the ex-
circles on (^) BC.CA.AS respect- ively.
r (^) radius of inscribed circle. R- —-—radius of circumscribed circle. rl> r2» r 3 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)
rl = (^) J8(s-a) (s-o) (s-c) s-a
r 2 - ^(a-aHa-bHa^T)
s-b r 3 " ^O^aMs-b) (a-c) 8-C
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.
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. 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.
2 2 2 2 To Prove: (^) AS 135 « 2 IE * 2 BT when (^) L is aid point
of BC.
Drew AD perpendicular to BC* Let L be the mid point of BC. In triangle (^) ABL
2 2 2 jar * ir be - (^2) <bd (dl) In triangle ALC 2 2 2 17 « IT (^) LTT 2 (cl) (dl)
But BL »^ LC. Adding equations (^2 2 2 ) H" IE »2aT 2BT also 2 2 2 2 Helc" «2aT (^) * Iff
Fig. 7.
In like nanner Ob) la^ antiparallel^ to^ AB FD is antlparallol^ to^ AC The antlparallelo ere parallel^ to^ the^ tangents^ at A, B and C. Since the radius of circumscribed circle Is perpendi- cular to the tangents then the radius la perpendicular to each antiparallel. .'. the lines joining the clrcuncentre to the vortices of a triangle are perpendicular to^ the^ sides of the pedal or orthocentric triangle*
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