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Triangle Geometry Hojoo Lee In this short note, we give some well-known formulae in Triangle Geomtry : 1. Law of Sines

b c a = = = 2R sinA sinB sinC

2. Law of Cosines a = bcosC + ccosB b2 + c2 − a2 cosA = 2bc

3. Area S, semiperimeter s, x = s − a, y = s − b, z = s − c p S = s(s − a)(s − b)(s − c) p = xyz(x + y + z) s X X 1 a4 a2 b2 − 2 = 4 cyclic cyclic = = = = = 4. cos

A , 2

cos

sin

A = 2

A , 2

r

tan

1 1 1 bcsinA = casinB = absinC 2 2 2 2 2R sinAsinBsinC rs = (s − a)rA = (s − b)rB = (s − c)rC √ rrA rB rC abc 4R

A 2

A s(s − a) , sin = bc 2

r

s A (s − b)(s − c) , tan = bc 2

r (s − b)(s − c) = s(s − a) s−a

5. R, r, rA , rB , rC 1 1 1 1 = + + r rA rB rC xyz (x + y)2 (y + z)2 (z + x)2 r2 = , R2 = x+y+z 16xyz(x + y + z) xyz A B C r = , r = 4Rsin sin sin R 2(x + y)(y + z)(z + x) 2 2 2 r abc (x + y)(y + z)(z + x) 1 + = cosA + cosB + cosC, Rr = = R 4s 2(x + y + z) 4R + r = rA + rB + rC ,

6. O(circumcenter), G(centroid), H(orthocenter), I(incenter) ~ = 3OG ~ = OA ~ + OB ~ + OC ~ Euler Line OGH : OH 1 X 1 X XA2 − BC 2 3 cyclic 9 cyclic ¡ ¢ ¡ ¢ 3 XA2 + XB 2 + XC 2 ≥ BC 2 + CA2 + AB 2 3(GA2 + GB 2 + GC 2 ) = BC 2 + CA2 + AB 2 4 HG2 = 4R2 − (BC 2 + CA2 + AB 2 ) 9 OH 2 = 9R2 − (BC 2 + CA2 + AB 2 ) XG2 =

XI 2

X cyclic 2

a + abc = 2

X

aXA2

cyclic 2

aXA + bXB + cXC ≥ abc IA2 IB 2 IC 2 + + =1 bc ca ab Euler : OI 2 = R2 − 2rR X 1 ~ = ~ OI aOA a + b + c cyclic ! à X X 1 5 a2 − 6 ab IG2 = r2 + 36 cyclic cyclic v à u µ ¶2 ! b+c u a tbc 1 − AI = a+b+c b+c AH = 2RcosA ~ · IG ~ = − 2 r(R − 2r) IH 3

7. Trigonometric Identities x+y y+z z+x sin sin 2 2 2 x+y y+z z+x cosx + cosy + cosz − cos(x + y + z) = 4cos cos cos 2 2 2

sinx + siny + sinz − sin(x + y + z) = 4sin

X

A B C sinA = 4cos cos cos 2 2 2 cyclic X sin2A = 4sinAsinBsinC cyclic

X

A B C cosA = 1 + 4sin sin sin 2 2 2 cyclic X cos2A = −1 − 4cosAcosBcosC cyclic

X

cos2 A = 1 − 2cosAcosBcosC

cyclic

X

sin2 A = 2 + 2cosAcosBcosC

cyclic

X

tanA = tanAtanBtanC

cyclic

X

cotAcotB = 1

cyclic

X

cyclic

cot

A A B C = cot cot cot 2 2 2 2