数学 におけるケイシーの定理 （ケイシーのていり、英 : Casey's theorem ）または一般化トレミーの定理 は、アイルランドの数学者 ジョン・ケイシー （英語版 ） ユークリッド幾何学 の定理である。
t
12
⋅
t
34
+
t
14
⋅
t
23
−
t
13
⋅
t
24
=
0
{\displaystyle t_{12}\cdot t_{34}+t_{14}\cdot t_{23}-t_{13}\cdot t_{24}=0}
O
{\displaystyle \,O}
R
{\displaystyle \,R}
O
1
,
O
2
,
O
3
,
O
4
{\displaystyle \,O_{1},O_{2},O_{3},O_{4}}
O
{\displaystyle \,O}
O
i
,
O
j
{\displaystyle \,O_{i},O_{j}}
接線 を引いたときの2接点の距離を
t
i
j
{\displaystyle \,t_{ij}}
[1]
t
12
⋅
t
34
+
t
14
⋅
t
23
=
t
13
⋅
t
24
{\displaystyle \,t_{12}\cdot t_{34}+t_{14}\cdot t_{23}=t_{13}\cdot t_{24}}
4つの円がみな1点にまで退化した場合、これはちょうどトレミーの定理になる。
以下の証明は Zacharias に帰せられる[2] [3]
O
i
{\displaystyle \,O_{i}}
R
i
{\displaystyle \,R_{i}}
O
{\displaystyle \,O}
K
i
{\displaystyle \,K_{i}}
O
,
O
i
{\displaystyle \,O,O_{i}}
ピタゴラスの定理 より、
t
i
j
2
=
O
i
O
j
¯
2
−
(
R
i
−
R
j
)
2
{\displaystyle \,t_{ij}^{2}={\overline {O_{i}O_{j}}}^{2}-(R_{i}-R_{j})^{2}}
この長さを点
K
i
,
K
j
{\displaystyle \,K_{i},K_{j}}
余弦定理 を三角形
O
i
O
O
j
{\displaystyle \,O_{i}OO_{j}}
O
i
O
j
¯
2
=
O
O
i
¯
2
+
O
O
j
¯
2
−
2
O
O
i
¯
⋅
O
O
j
¯
⋅
cos
∠
O
i
O
O
j
{\displaystyle {\overline {O_{i}O_{j}}}^{2}={\overline {OO_{i}}}^{2}+{\overline {OO_{j}}}^{2}-2{\overline {OO_{i}}}\cdot {\overline {OO_{j}}}\cdot \cos \angle O_{i}OO_{j}}
が得られる。円
O
,
O
i
{\displaystyle \,O,O_{i}}
O
O
i
¯
=
R
−
R
i
,
∠
O
i
O
O
j
=
∠
K
i
O
K
j
{\displaystyle {\overline {OO_{i}}}=R-R_{i},\,\angle O_{i}OO_{j}=\angle K_{i}OK_{j}}
C
{\displaystyle \,C}
O
{\displaystyle \,O}
正弦定理 を三角形
K
i
C
K
j
{\displaystyle \,K_{i}CK_{j}}
K
i
K
j
¯
=
2
R
⋅
sin
∠
K
i
C
K
j
=
2
R
⋅
sin
∠
K
i
O
K
j
2
{\displaystyle {\overline {K_{i}K_{j}}}=2R\cdot \sin \angle K_{i}CK_{j}=2R\cdot \sin {\frac {\angle K_{i}OK_{j}}{2}}}
が得られる。これより、
cos
∠
K
i
O
K
j
=
1
−
2
sin
2
∠
K
i
O
K
j
2
=
1
−
2
⋅
(
K
i
K
j
¯
2
R
)
2
=
1
−
K
i
K
j
¯
2
2
R
2
{\displaystyle \cos \angle K_{i}OK_{j}=1-2\sin ^{2}{\frac {\angle K_{i}OK_{j}}{2}}=1-2\cdot \left({\frac {\overline {K_{i}K_{j}}}{2R}}\right)^{2}=1-{\frac {{\overline {K_{i}K_{j}}}^{2}}{2R^{2}}}}
以上を余弦定理の式に代入すると
O
i
O
j
¯
2
=
(
R
−
R
i
)
2
+
(
R
−
R
j
)
2
−
2
(
R
−
R
i
)
(
R
−
R
j
)
(
1
−
K
i
K
j
¯
2
2
R
2
)
=
(
R
−
R
i
)
2
+
(
R
−
R
j
)
2
−
2
(
R
−
R
i
)
(
R
−
R
j
)
+
(
R
−
R
i
)
(
R
−
R
j
)
⋅
K
i
K
j
¯
2
R
2
=
(
(
R
−
R
i
)
−
(
R
−
R
j
)
)
2
+
(
R
−
R
i
)
(
R
−
R
j
)
⋅
K
i
K
j
¯
2
R
2
{\displaystyle {\begin{aligned}{\overline {O_{i}O_{j}}}^{2}&=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})\left(1-{\frac {{\overline {K_{i}K_{j}}}^{2}}{2R^{2}}}\right)\\&=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})+(R-R_{i})(R-R_{j})\cdot {\frac {{\overline {K_{i}K_{j}}}^{2}}{R^{2}}}\\&=((R-R_{i})-(R-R_{j}))^{2}+(R-R_{i})(R-R_{j})\cdot {\frac {{\overline {K_{i}K_{j}}}^{2}}{R^{2}}}\end{aligned}}}
よって
t
i
j
=
O
i
O
j
¯
2
−
(
R
i
−
R
j
)
2
=
R
−
R
i
⋅
R
−
R
j
⋅
K
i
K
j
¯
R
{\displaystyle t_{ij}={\sqrt {{\overline {O_{i}O_{j}}}^{2}-(R_{i}-R_{j})^{2}}}={\frac {{\sqrt {R-R_{i}}}\cdot {\sqrt {R-R_{j}}}\cdot {\overline {K_{i}K_{j}}}}{R}}}
が得られる。円に内接する四角形
K
1
K
2
K
3
K
4
{\displaystyle \,K_{1}K_{2}K_{3}K_{4}}
t
12
t
34
+
t
14
t
23
=
1
R
2
⋅
R
−
R
1
R
−
R
2
R
−
R
3
R
−
R
4
(
K
1
K
2
¯
⋅
K
3
K
4
¯
+
K
1
K
4
¯
⋅
K
2
K
3
¯
)
=
1
R
2
⋅
R
−
R
1
R
−
R
2
R
−
R
3
R
−
R
4
(
K
1
K
3
¯
⋅
K
2
K
4
¯
)
=
t
13
t
24
{\displaystyle {\begin{aligned}&t_{12}t_{34}+t_{14}t_{23}\\[4pt]={}&{\frac {1}{R^{2}}}\cdot {\sqrt {R-R_{1}}}{\sqrt {R-R_{2}}}{\sqrt {R-R_{3}}}{\sqrt {R-R_{4}}}\left({\overline {K_{1}K_{2}}}\cdot {\overline {K_{3}K_{4}}}+{\overline {K_{1}K_{4}}}\cdot {\overline {K_{2}K_{3}}}\right)\\[4pt]={}&{\frac {1}{R^{2}}}\cdot {\sqrt {R-R_{1}}}{\sqrt {R-R_{2}}}{\sqrt {R-R_{3}}}{\sqrt {R-R_{4}}}\left({\overline {K_{1}K_{3}}}\cdot {\overline {K_{2}K_{4}}}\right)\\[4pt]={}&t_{13}t_{24}\end{aligned}}}
4つの円が最大の円に内側から接していなくともよい。実際、これらが外側から接している場合も考えることができて、その場合は以下のように定めればよい[4]
円
O
i
,
O
j
{\displaystyle \,O_{i},O_{j}}
O
{\displaystyle \,O}
t
i
j
{\displaystyle \,t_{ij}}
円
O
i
,
O
j
{\displaystyle \,O_{i},O_{j}}
O
{\displaystyle \,O}
t
i
j
{\displaystyle \,t_{ij}}
ケイシーの定理の逆もまた成り立つ[4]
^ a b
Casey, J. (1866). “On the Equations and Properties: (1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane”. Proceedings of the Royal Irish Academy 9 : 396–423. JSTOR 20488927 .
^
Bottema, O. (1944). Hoofdstukken uit de Elementaire Meetkunde . (translation by Reinie Erné as Topics in Elementary Geometry , Springer 2008, of the second extended edition published by Epsilon-Uitgaven 1987)
^
Zacharias, M. (1942). “Der Caseysche Satz”. Jahresbericht der Deutschen Mathematiker-Vereinigung 52 : 79–89.
^ a b
Johnson, Roger A. (1929). Modern Geometry . Houghton Mifflin, Boston (republished facsimile by Dover 1960, 2007 as Advanced Euclidean Geometry )
ウィキメディア・コモンズには、
ケイシーの定理 に関連するカテゴリがあります。
![{\displaystyle {\begin{aligned}&t_{12}t_{34}+t_{14}t_{23}\\[4pt]={}&{\frac {1}{R^{2}}}\cdot {\sqrt {R-R_{1}}}{\sqrt {R-R_{2}}}{\sqrt {R-R_{3}}}{\sqrt {R-R_{4}}}\left({\overline {K_{1}K_{2}}}\cdot {\overline {K_{3}K_{4}}}+{\overline {K_{1}K_{4}}}\cdot {\overline {K_{2}K_{3}}}\right)\\[4pt]={}&{\frac {1}{R^{2}}}\cdot {\sqrt {R-R_{1}}}{\sqrt {R-R_{2}}}{\sqrt {R-R_{3}}}{\sqrt {R-R_{4}}}\left({\overline {K_{1}K_{3}}}\cdot {\overline {K_{2}K_{4}}}\right)\\[4pt]={}&t_{13}t_{24}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3293ace55a8f18b5d3bacff95333c091f91ac1d)
