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Question Number 196829 by cortano12 last updated on 01/Sep/23

	Answered by universe last updated on 01/Sep/23

	$b y a p o l l o n i u s t h e o r e m$ ${B C}^{2} + {P C}^{2} = 2 ({Q C}^{2} + {P Q}^{2}) . . . . . (1)$ ${A C}^{2} + {Q C}^{2} = 2 ({P C}^{2} + {P Q}^{2}) . . . . . (2)$ $(1) + (2)$ ${A C}^{2} + {B C}^{2} + {P C}^{2} + {Q C}^{2} = 2 {Q C}^{2} + 2 {P C}^{2} + 4 {P Q}^{2}$ ${A B}^{2} = 3 {P Q}^{2} + {P Q}^{2} + {Q C}^{2} + {P C}^{2}$ ${A B}^{2} - 3 {P Q}^{2} = {P Q}^{2} + {P C}^{2} + {Q C}^{2}$ $A B = 3 P Q$ ${A B}^{2} - \frac{3 {A B}^{2}}{9} = {P Q}^{2} + {P C}^{2} + {Q C}^{2}$ $\frac{2}{3} {A B}^{2} = {P Q}^{2} + {Q C}^{2} + {P C}^{2}$
	Commented by cortano12 last updated on 01/Sep/23

	${AB}^{2} - \frac{{3 AB}^{2}}{9} = \frac{2}{3} {AB}^{2}$
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