Home
Random
Nearby
Log in
Settings
Donate
About Wikipedia
Disclaimers
Search
User
:
Fjackson/workinprogress
User page
Talk
Language
Watch
Edit
<
User:Fjackson
Useful formulas in
Conway triangle notation
:
∑
cyclic
1
a
2
=
1
4
R
2
S
2
(
S
2
+
S
ω
2
)
{\displaystyle \sum _{\text{cyclic}}{\frac {1}{a^{2}}}={\frac {1}{4R^{2}S^{2}}}(S^{2}+S_{\omega }^{2})\,}
∑
cyclic
1
a
=
1
4
r
2
R
S
(
S
2
−
2
r
2
S
ω
)
{\displaystyle \sum _{\text{cyclic}}{\frac {1}{a}}={\frac {1}{4r^{2}RS}}(S^{2}-2r^{2}S_{\omega })\,}
where r is the inradius
∑
cyclic
a
=
S
r
{\displaystyle \sum _{\text{cyclic}}a={\frac {S}{r}}\,}
∑
cyclic
a
2
=
2
S
ω
{\displaystyle \sum _{\text{cyclic}}a^{2}=2S_{\omega }\,}
∑
cyclic
a
3
=
2
S
r
3
(
12
R
r
3
+
6
S
ω
r
2
−
S
2
)
{\displaystyle \sum _{\text{cyclic}}a^{3}={\frac {2S}{r^{3}}}(12Rr^{3}+6S_{\omega }r^{2}-S^{2})\,}
∑
cyclic
a
4
=
2
(
S
ω
2
−
S
2
)
{\displaystyle \sum _{\text{cyclic}}a^{4}=2(S_{\omega }^{2}-S^{2})\,}
∑
cyclic
a
6
=
2
(
S
ω
3
−
3
S
ω
S
2
+
6
R
2
S
2
)
{\displaystyle \sum _{\text{cyclic}}a^{6}=2(S_{\omega }^{3}-3S_{\omega }S^{2}+6R^{2}S^{2})\,}
∑
cyclic
1
S
A
2
=
(
S
2
−
2
S
ω
2
+
8
R
2
S
ω
)
S
2
(
S
ω
−
4
R
2
)
{\displaystyle \sum _{\text{cyclic}}{\frac {1}{S_{A}^{2}}}={\frac {(S^{2}-2S_{\omega }^{2}+8R^{2}S_{\omega })}{S^{2}(S_{\omega }-4R^{2})}}\,}
∑
cyclic
1
S
A
=
1
S
ω
−
4
R
2
{\displaystyle \sum _{\text{cyclic}}{\frac {1}{S_{A}}}={\frac {1}{S_{\omega }-4R^{2}}}\,}
∑
cyclic
S
A
=
S
ω
{\displaystyle \sum _{\text{cyclic}}S_{A}=S_{\omega }\,}
∑
cyclic
S
A
2
=
S
ω
2
−
2
S
2
{\displaystyle \sum _{\text{cyclic}}S_{A}^{2}=S_{\omega }^{2}-2S^{2}\,}
∑
cyclic
S
A
3
=
S
ω
3
−
12
R
2
S
2
{\displaystyle \sum _{\text{cyclic}}S_{A}^{3}=S_{\omega }^{3}-12R^{2}S^{2}\,}
∑
cyclic
S
A
4
=
S
ω
4
+
2
S
2
+
16
R
2
S
ω
S
2
{\displaystyle \sum _{\text{cyclic}}S_{A}^{4}=S_{\omega }^{4}+2S^{2}+16R^{2}S_{\omega }S^{2}\,}
∑
cyclic
1
b
2
c
2
=
S
ω
2
S
2
R
2
{\displaystyle \sum _{\text{cyclic}}{\frac {1}{b^{2}c^{2}}}={\frac {S_{\omega }}{2S^{2}R^{2}}}\,}
∑
cyclic
1
b
c
=
1
2
r
R
{\displaystyle \sum _{\text{cyclic}}{\frac {1}{bc}}={\frac {1}{2rR}}\,}
∑
cyclic
b
c
=
1
2
r
2
(
S
2
−
2
r
2
S
ω
)
{\displaystyle \sum _{\text{cyclic}}bc={\frac {1}{2r^{2}}}(S^{2}-2r^{2}S_{\omega })\,}
∑
cyclic
b
2
c
2
=
S
ω
2
+
S
2
{\displaystyle \sum _{\text{cyclic}}b^{2}c^{2}=S_{\omega }^{2}+S^{2}\,}