カテゴリー: 幾何学

\[ \cos A=\frac{\left|DA\right|^{2}+\left|AB\right|^{2}-\left|BC\right|^{2}-\left|CD\right|^{2}}{2\left(\left|DA\right|\left|AB\right|+\left|BC\right|\left|CD\right|\right)} \]

\[ \text{3角形の3辺の長さが}a,b,c\Leftrightarrow\left|b-c\right|<a<b+c \]

$0^{\circ}$より大きく$90^{\circ}$より小さい角を鋭角という。

\[ A<B\Leftrightarrow a<b \]

\[ \sin A+\sin B+\sin C=4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} \]

\[ a\cos A+b\cos B+c\cos C=\frac{8S^{2}}{abc} \]

\[ \left|AG\right|^{2}=\frac{-a^{2}+2b^{2}+2c^{2}}{9} \]

\[ \boldsymbol{H}+2\boldsymbol{J}=3\boldsymbol{G} \]

\[ a^{2}=b^{2}+c^{2}-2bc\cos A \]

\[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R \]

\[ r_{a}=\frac{S}{s-a} \]

\begin{align*} S & =\frac{abc}{4R}\\ & =\frac{1}{2}r\left(a+b+c\right)\\ & =2R^{2}\sin A\sin B\sin C\\ & =rR\left(\sin A+\sin B+\sin C\right) \end{align*}

\[ \boldsymbol{H}=\frac{\tan A\boldsymbol{A}+\tan B\boldsymbol{B}+\tan C\boldsymbol{C}}{\tan A\tan B\tan C} \]

\[ r=\frac{S}{s} \]

\[ \boldsymbol{X}=\frac{p\boldsymbol{A}+q\boldsymbol{B}+r\boldsymbol{C}}{p+q+r} \]

\[ S=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \]

\[ S=\sqrt{abcd}\sin\frac{A+C}{2} \]

\[ S=\sqrt{\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)} \]

\[ \left|\overrightarrow{AB}\right|+\left|\overrightarrow{CD}\right|=\left|\overrightarrow{BC}\right|+\left|\overrightarrow{DA}\right| \]

\[ S=\sqrt{\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)-abcd\cos^{2}\frac{A+C}{2}} \]

\[ S=\frac{1}{2}\left(\overrightarrow{AC}\times\overrightarrow{DB}\right) \]

\[ \left|\overrightarrow{AB}\right|\left|\overrightarrow{CD}\right|+\left|\overrightarrow{BC}\right|\left|\overrightarrow{DA}\right|=\left|\overrightarrow{BD}\right|\left|\overrightarrow{CA}\right| \]

\[ p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos\left(A+C\right) \]

\[ \overrightarrow{AB}\cdot\overrightarrow{CD}=\frac{1}{2}\left(b^{2}+d^{2}-p^{2}-q^{2}\right) \]

\[ x^{2}+y^{2}-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{\;2}+y_{1}^{\;2}\\ x_{2}^{\;2}+y_{2}^{\;2}\\ x_{3}^{\;2}+y_{3}^{\;2} \end{array}\right)=0 \]

\[ \frac{a^{2}+b^{2}}{4}-c>0 \]