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Związek między długościami boków trójkąta a długościami jego środkowych

związek-między-długościami-boków-trójkąta-a-długościami-jego-środkowych-7ed755ac-6e3d-40f1-b3ab-c75b83e175fe

Związek między długościami boków trójkąta a długościami jego środkowych

a = 2 3 2 m b 2 + 2 m c 2 m a 2 b = 2 3 2 m c 2 + 2 m a 2 m b 2 c = 2 3 2 m b 2 + 2 m a 2 m c 2 a = 2 3 2 m b 2 + 2 m c 2 m a 2 b = 2 3 2 m c 2 + 2 m a 2 m b 2 c = 2 3 2 m b 2 + 2 m a 2 m c 2 {:[a=(2)/(3)sqrt(2m_(b)^(2)+2m_(c)^(2)-m_(a)^(2))],[b=(2)/(3)sqrt(2m_(c)^(2)+2m_(a)^(2)-m_(b)^(2))],[c=(2)/(3)sqrt(2m_(b)^(2)+2m_(a)^(2)-m_(c)^(2))]:}\begin{aligned} &a=\frac{2}{3} \sqrt{2 m_{b}^{2}+2 m_{c}^{2}-m_{a}^{2}} \\ &b=\frac{2}{3} \sqrt{2 m_{c}^{2}+2 m_{a}^{2}-m_{b}^{2}} \\ &c=\frac{2}{3} \sqrt{2 m_{b}^{2}+2 m_{a}^{2}-m_{c}^{2}} \end{aligned}a=232mb2+2mc2ma2b=232mc2+2ma2mb2c=232mb2+2ma2mc2


Dowód:

Z E B C Z E B C Z/_\EBC\mathrm{Z} \triangle E B CZEBC i tw. kosinusów:
a 2 = m c 2 + ( 1 2 c ) 2 2 m c 1 2 c cos α a 2 = m c 2 + 1 4 c 2 m c c cos α a 2 = m c 2 + 1 2 c 2 2 m c 1 2 c cos α a 2 = m c 2 + 1 4 c 2 m c c cos α a^(2)=m_(c)^(2)+((1)/(2)c)^(2)-2m_(c)(1)/(2)c cos alpha=>a^(2)=m_(c)^(2)+(1)/(4)c^(2)-m_(c)c cos alphaa^{2}=m_{c}^{2}+\left(\frac{1}{2} c\right)^{2}-2 m_{c} \frac{1}{2} c \cos \alpha \Rightarrow a^{2}=m_{c}^{2}+\frac{1}{4} c^{2}-m_{c} c \cos \alphaa2=mc2+(12c)22mc12ccosαa2=mc2+14c2mcccosα
Z E B S E B S /_\EBS\triangle E B SEBS i tw. kosinusów:
( 2 3 m b ) 2 = ( 1 3 m c ) 2 + ( 1 2 c ) 2 2 1 3 m c 1 2 c cos α 2 3 m b 2 = 1 3 m c 2 + 1 2 c 2 2 1 3 m c 1 2 c cos α ((2)/(3)m_(b))^(2)=((1)/(3)m_(c))^(2)+((1)/(2)c)^(2)-2*(1)/(3)m_(c)(1)/(2)c cos alpha\left(\frac{2}{3} m_{b}\right)^{2}=\left(\frac{1}{3} m_{c}\right)^{2}+\left(\frac{1}{2} c\right)^{2}-2 \cdot \frac{1}{3} m_{c} \frac{1}{2} c \cos \alpha(23mb)2=(13mc)2+(12c)2213mc12ccosα
Z A E S A E S /_\AES\triangle A E SAES i tw. kosinusów:
( 2 3 m a ) 2 = ( 1 3 m c ) 2 + ( 1 2 c ) 2 2 1 3 m c 1 2 c cos ( 180 α ) ( 2 3 m a ) 2 = ( 1 3 m c ) 2 + ( 1 2 c ) 2 + 2 1 3 m c 1 2 c cos α 2 3 m a 2 = 1 3 m c 2 + 1 2 c 2 2 1 3 m c 1 2 c cos 180 α 2 3 m a 2 = 1 3 m c 2 + 1 2 c 2 + 2 1 3 m c 1 2 c cos α {:[((2)/(3)m_(a))^(2)=((1)/(3)m_(c))^(2)+((1)/(2)c)^(2)-2*(1)/(3)m_(c)*(1)/(2)c cos(180^(@)-alpha)],[((2)/(3)m_(a))^(2)=((1)/(3)m_(c))^(2)+((1)/(2)c)^(2)+2*(1)/(3)m_(c)*(1)/(2)c cos alpha]:}\begin{aligned} &\left(\frac{2}{3} m_{a}\right)^{2}=\left(\frac{1}{3} m_{c}\right)^{2}+\left(\frac{1}{2} c\right)^{2}-2 \cdot \frac{1}{3} m_{c} \cdot \frac{1}{2} c \cos \left(180^{\circ}-\alpha\right) \\ &\left(\frac{2}{3} m_{a}\right)^{2}=\left(\frac{1}{3} m_{c}\right)^{2}+\left(\frac{1}{2} c\right)^{2}+2 \cdot \frac{1}{3} m_{c} \cdot \frac{1}{2} c \cos \alpha \end{aligned}(23ma)2=(13mc)2+(12c)2213mc12ccos(180α)(23ma)2=(13mc)2+(12c)2+213mc12ccosα
{ ( 2 3 m b ) 2 = ( 1 3 m c ) 2 + ( 1 2 c ) 2 1 3 m c c cos α + ( 2 3 m a ) 2 = ( 1 3 m c ) 2 + ( 1 2 c ) 2 + 1 3 m c c cos α ( 2 3 m b ) 2 + ( 2 3 m a ) 2 = 2 ( 1 3 m c ) 2 + 2 ( 1 2 c ) 2 4 9 m b 2 + 4 9 m a 2 = 2 g m c 2 + 1 2 c 2 1 2 c 2 = 4 9 m b 2 + 4 9 m a 2 2 9 m c 2 / 2 c 2 = 8 9 m b 2 + 8 9 m a 2 4 9 m c 2 c 2 = 4 9 ( 2 m b 2 + 2 m a 2 m c 2 ) c = 2 3 2 m b 2 + 2 m a 2 m c 2 2 3 m b 2 = 1 3 m c 2 + 1 2 c 2 1 3 m c c cos α + 2 3 m a 2 = 1 3 m c 2 + 1 2 c 2 + 1 3 m c c cos α 2 3 m b 2 + 2 3 m a 2 = 2 1 3 m c 2 + 2 1 2 c 2 4 9 m b 2 + 4 9 m a 2 = 2 g m c 2 + 1 2 c 2 1 2 c 2 = 4 9 m b 2 + 4 9 m a 2 2 9 m c 2 / 2 c 2 = 8 9 m b 2 + 8 9 m a 2 4 9 m c 2 c 2 = 4 9 2 m b 2 + 2 m a 2 m c 2 c = 2 3 2 m b 2 + 2 m a 2 m c 2 {:[{[((2)/(3)m_(b))^(2)=((1)/(3)m_(c))^(2)+((1)/(2)c)^(2)-(1)/(3)m_(c)c cos alpha],[+((2)/(3)m_(a))^(2)=((1)/(3)m_(c))^(2)+((1)/(2)c)^(2)+(1)/(3)m_(c)c cos alpha]:}],[((2)/(3)m_(b))^(2)+((2)/(3)m_(a))^(2)=2((1)/(3)m_(c))^(2)+2*((1)/(2)c)^(2)],[(4)/(9)m_(b)^(2)+(4)/(9)m_(a)^(2)=(2)/(g)m_(c)^(2)+(1)/(2)c^(2)=>(1)/(2)c^(2)=(4)/(9)m_(b)^(2)+(4)/(9)m_(a)^(2)-(2)/(9)m_(c)^(2)//*2=>c^(2)=(8)/(9)m_(b)^(2)+(8)/(9)m_(a)^(2)-(4)/(9)m_(c)^(2)=>],[=>c^(2)=(4)/(9)(2m_(b)^(2)+2m_(a)^(2)-m_(c)^(2))=>c=(2)/(3)sqrt(2m_(b)^(2)+2m_(a)^(2)-m_(c)^(2))]:}\begin{aligned} & \left\{\begin{array}{l}\left(\frac{2}{3} m_{b}\right)^{2}=\left(\frac{1}{3} m_{c}\right)^{2}+\left(\frac{1}{2} c\right)^{2}-\frac{1}{3} m_{c} c \cos \alpha \\+\left(\frac{2}{3} m_{a}\right)^{2}=\left(\frac{1}{3} m_{c}\right)^{2}+\left(\frac{1}{2} c\right)^{2}+\frac{1}{3} m_{c} c \cos \alpha\end{array}\right. \\ & \left(\frac{2}{3} m_{b}\right)^{2}+\left(\frac{2}{3} m_{a}\right)^{2}=2\left(\frac{1}{3} m_{c}\right)^{2}+2 \cdot\left(\frac{1}{2} c\right)^{2} \\ & \frac{4}{9} m_{b}^{2}+\frac{4}{9} m_{a}^{2}=\frac{2}{g} m_{c}^{2}+\frac{1}{2} c^{2} \Rightarrow \frac{1}{2} c^{2}=\frac{4}{9} m_{b}^{2}+\frac{4}{9} m_{a}^{2}-\frac{2}{9} m_{c}^{2} / \cdot 2 \Rightarrow c^{2}=\frac{8}{9} m_{b}^{2}+\frac{8}{9} m_{a}^{2}-\frac{4}{9} m_{c}^{2} \Rightarrow \\ & \Rightarrow c^{2}=\frac{4}{9}\left(2 m_{b}^{2}+2 m_{a}^{2}-m_{c}^{2}\right) \Rightarrow c=\frac{2}{3} \sqrt{2 m_{b}^{2}+2 m_{a}^{2}-m_{c}^{2}} \end{aligned}{(23mb)2=(13mc)2+(12c)213mcccosα+(23ma)2=(13mc)2+(12c)2+13mcccosα(23mb)2+(23ma)2=2(13mc)2+2(12c)249mb2+49ma2=2gmc2+12c212c2=49mb2+49ma229mc2/2c2=89mb2+89ma249mc2c2=49(2mb2+2ma2mc2)c=232mb2+2ma2mc2
Analogicznie wyprowadzamy wzory dla boku a a aaa i b b bbb.
Zatem boki trójkąta można wyrazić za pomocą środkowych trójkąta w następujący sposób
a = 2 3 2 m b 2 + 2 m c 2 m a 2 b = 2 3 2 m c 2 + 2 m a 2 m b 2 c = 2 3 2 m b 2 + 2 m a 2 m c 2 a = 2 3 2 m b 2 + 2 m c 2 m a 2 b = 2 3 2 m c 2 + 2 m a 2 m b 2 c = 2 3 2 m b 2 + 2 m a 2 m c 2 {:[a=(2)/(3)sqrt(2m_(b)^(2)+2m_(c)^(2)-m_(a)^(2))],[b=(2)/(3)sqrt(2m_(c)^(2)+2m_(a)^(2)-m_(b)^(2))],[c=(2)/(3)sqrt(2m_(b)^(2)+2m_(a)^(2)-m_(c)^(2))]:}\begin{aligned} &a=\frac{2}{3} \sqrt{2 m_{b}^{2}+2 m_{c}^{2}-m_{a}^{2}} \\ &b=\frac{2}{3} \sqrt{2 m_{c}^{2}+2 m_{a}^{2}-m_{b}^{2}} \\ &c=\frac{2}{3} \sqrt{2 m_{b}^{2}+2 m_{a}^{2}-m_{c}^{2}} \end{aligned}a=232mb2+2mc2ma2b=232mc2+2ma2mb2c=232mb2+2ma2mc2


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