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Question Number 207533 by mr W last updated on 18/May/24

Commented by mr W last updated on 18/May/24

$f i n d t h e r a d i u s o f c i r c u m c i r c l e$
	Answered by mr W last updated on 20/May/24
	$OA=p=R−a OB=q=R−b OC=r=R−c X=q^2 +r^2 −u^2 =(R−b)^2 +(R−c)^2 −u^2 =2R^2 −2(b+c)R+b^2 +c^2 −u^2 Y=r^2 +p^2 −v^2 =(R−c)^2 +(R−a)^2 −v^2 =2R^2 −2(c+a)R+c^2 +a^2 −v^2 Z=p^2 +q^2 −w^2 =(R−a)^2 +(R−b)^2 −w^2 =2R^2 −2(a+b)R+a^2 +b^2 −w^2 volume of tetrahedron O−ABC should be zero. V=((√(4p^2 q^2 r^2 +XYZ−p^2 X^2 −q^2 Y^2 −r^2 Z^2 ))/(12))=0 4p^2 q^2 r^2 +XYZ−p^2 X^2 −q^2 Y^2 −r^2 Z^2 =0 4(R−a)^2 (R−b)^2 (R−c)^2 +XYZ−(R−a)^2 X^2 −(R−b)^2 Y^2 −(R−c)^2 Z^2 =0 after expansion we get a quadratic equation for R: {−(u^4 +v^4 +w^4 )+2(u^2 v^2 +v^2 w^2 +w^2 u^2 )+4[a(b+c)−bc−a^2 ]u^2 +4[b(c+a)−ca−b^2 ]v^2 +4[c(a+b)−ab−c^2 ]w^2 }R^2 +2{au^4 +bv^4 +cw^4 +2(a^3 u^2 +b^3 v^2 +c^3 w^2 )−(a+b)(u^2 v^2 +c^2 w^2 )−(b+c)(v^2 w^2 +a^2 u^2 )−(c+a)(w^2 u^2 +b^2 v^2 )+[bc(b+c)−a(b^2 +c^2 )]u^2 +[ca(c+a)−b(c^2 +a^2 )]v^2 +[ab(a+b)−c(a^2 +b^2 )]w^2 }R −{a^2 u^2 (a^2 +u^2 )+b^2 v^2 (b^2 +v^2 )+c^2 w^2 (c^2 +w^2 )−[a^2 (b^2 +c^2 )−b^2 c^2 ]u^2 −[b^2 (c^2 +a^2 )−c^2 a^2 ]v^2 −[c^2 (a^2 +b^2 )−a^2 b^2 ]w^2 −(a^2 +b^2 )u^2 v^2 −(b^2 +c^2 )v^2 w^2 −(c^2 +a^2 )w^2 u^2 +u^2 v^2 w^2 }=0 with A=−(u^4 +v^4 +w^4 )+2(u^2 v^2 +v^2 w^2 +w^2 u^2 )+4[a(b+c)−bc−a^2 ]u^2 +4[b(c+a)−ca−b^2 ]v^2 +4[c(a+b)−ab−c^2 ]w^2 B=au^4 +bv^4 +cw^4 +2(a^3 u^2 +b^3 v^2 +c^3 w^2 )−(a+b)(u^2 v^2 +c^2 w^2 )−(b+c)(v^2 w^2 +a^2 u^2 )−(c+a)(w^2 u^2 +b^2 v^2 )+[bc(b+c)−a(b^2 +c^2 )]u^2 +[ca(c+a)−b(c^2 +a^2 )]v^2 +[ab(a+b)−c(a^2 +b^2 )]w^2 C=a^2 u^2 (a^2 +u^2 )+b^2 v^2 (b^2 +v^2 )+c^2 w^2 (c^2 +w^2 )−[a^2 (b^2 +c^2 )−b^2 c^2 ]u^2 −[b^2 (c^2 +a^2 )−c^2 a^2 ]v^2 −[c^2 (a^2 +b^2 )−a^2 b^2 ]w^2 −(a^2 +b^2 )u^2 v^2 −(b^2 +c^2 )v^2 w^2 −(c^2 +a^2 )w^2 u^2 +u^2 v^2 w^2 ⇒AR^2 +2BR−C=0 ⇒R=((−B±(√(B^2 +AC)))/A) example: u=7, v=6, w=8 a=3, b=2, c=1 A=−(7^4 +6^4 +8^4 )+2(7^2 6^2 +6^2 8^2 +8^2 7^2 )+4[3(2+1)−2×1−3^2 ]7^2 +4[2(1+3)−1×3−2^2 ]6^2 +4[1(3+2)−3×2−1^2 ]8^2 =5855 B=3×7^4 +2×6^4 +1×8^4 +2(3^3 7^2 +2^3 6^2 +1^3 8^2 )−(3+2)(7^2 6^2 +1^2 8^2 )−(2+1)(6^2 8^2 +3^2 7^2 )−(1+3)(8^2 7^2 +2^2 6^2 )+[2×1(2+1)−3(2^2 +1^2 )]7^2 +[1×3(1+3)−2(1^2 +3^2 )]6^2 +[3×2(3+2)−1(3^2 +2^2 )]8^2 =−12895 C=3^2 7^2 (3^2 +7^2 )+2^2 6^2 (2^2 +6^2 )+1^2 8^2 (1^2 +8^2 )−[3^2 (2^2 +1^2 )−2^2 1^2 ]7^2 −[2^2 (1^2 +3^2 )−1^2 3^2 ]6^2 −[1^2 (3^2 +2^2 )−3^2 2^2 ]8^2 −(3^2 +2^2 )7^2 6^2 −(2^2 +1^2 )6^2 8^2 −(1^2 +3^2 )8^2 7^2 +7^2 6^2 8^2 =80929 R=((12895±(√(12895^2 +5855×80929)))/(5855)) =((2579)/(1171))±((3024(√(70)))/(5855)) ≈6.523586 / −2.118804$
	$O A = p = R - a$ $O B = q = R - b$ $O C = r = R - c$ $X = q^{2} + r^{2} - u^{2} = {(R - b)}^{2} + {(R - c)}^{2} - u^{2} = 2 R^{2} - 2 (b + c) R + b^{2} + c^{2} - u^{2}$ $Y = r^{2} + p^{2} - v^{2} = {(R - c)}^{2} + {(R - a)}^{2} - v^{2} = 2 R^{2} - 2 (c + a) R + c^{2} + a^{2} - v^{2}$ $Z = p^{2} + q^{2} - w^{2} = {(R - a)}^{2} + {(R - b)}^{2} - w^{2} = 2 R^{2} - 2 (a + b) R + a^{2} + b^{2} - w^{2}$ $v o l u m e o f t e t r a h e d r o n O - A B C$ $s h o u l d b e z e r o .$ $V = \frac{\sqrt{4 p^{2} q^{2} r^{2} + X Y Z - p^{2} X^{2} - q^{2} Y^{2} - r^{2} Z^{2}}}{12} = 0$ $4 p^{2} q^{2} r^{2} + X Y Z - p^{2} X^{2} - q^{2} Y^{2} - r^{2} Z^{2} = 0$ $4 {(R - a)}^{2} {(R - b)}^{2} {(R - c)}^{2} + X Y Z - {(R - a)}^{2} X^{2} - {(R - b)}^{2} Y^{2} - {(R - c)}^{2} Z^{2} = 0$ $a f t e r e x p a n s i o n w e g e t a q u a d r a t i c$ $e q u a t i o n f o r R :$ ${- (u^{4} + v^{4} + w^{4}) + 2 (u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}) + 4 [a (b + c) - b c - a^{2}] u^{2} + 4 [b (c + a) - c a - b^{2}] v^{2} + 4 [c (a + b) - a b - c^{2}] w^{2}} R^{2}$ $+ 2 {{a u}^{4} + {b v}^{4} + {c w}^{4} + 2 (a^{3} u^{2} + b^{3} v^{2} + c^{3} w^{2}) - (a + b) (u^{2} v^{2} + c^{2} w^{2}) - (b + c) (v^{2} w^{2} + a^{2} u^{2}) - (c + a) (w^{2} u^{2} + b^{2} v^{2}) + [b c (b + c) - a (b^{2} + c^{2})] u^{2} + [c a (c + a) - b (c^{2} + a^{2})] v^{2} + [a b (a + b) - c (a^{2} + b^{2})] w^{2}} R$ $- {a^{2} u^{2} (a^{2} + u^{2}) + b^{2} v^{2} (b^{2} + v^{2}) + c^{2} w^{2} (c^{2} + w^{2}) - [a^{2} (b^{2} + c^{2}) - b^{2} c^{2}] u^{2} - [b^{2} (c^{2} + a^{2}) - c^{2} a^{2}] v^{2} - [c^{2} (a^{2} + b^{2}) - a^{2} b^{2}] w^{2} - (a^{2} + b^{2}) u^{2} v^{2} - (b^{2} + c^{2}) v^{2} w^{2} - (c^{2} + a^{2}) w^{2} u^{2} + u^{2} v^{2} w^{2}} = 0$ $w i t h$ $A = - (u^{4} + v^{4} + w^{4}) + 2 (u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}) + 4 [a (b + c) - b c - a^{2}] u^{2} + 4 [b (c + a) - c a - b^{2}] v^{2} + 4 [c (a + b) - a b - c^{2}] w^{2}$ $B = {a u}^{4} + {b v}^{4} + {c w}^{4} + 2 (a^{3} u^{2} + b^{3} v^{2} + c^{3} w^{2}) - (a + b) (u^{2} v^{2} + c^{2} w^{2}) - (b + c) (v^{2} w^{2} + a^{2} u^{2}) - (c + a) (w^{2} u^{2} + b^{2} v^{2}) + [b c (b + c) - a (b^{2} + c^{2})] u^{2} + [c a (c + a) - b (c^{2} + a^{2})] v^{2} + [a b (a + b) - c (a^{2} + b^{2})] w^{2}$ $C = a^{2} u^{2} (a^{2} + u^{2}) + b^{2} v^{2} (b^{2} + v^{2}) + c^{2} w^{2} (c^{2} + w^{2}) - [a^{2} (b^{2} + c^{2}) - b^{2} c^{2}] u^{2} - [b^{2} (c^{2} + a^{2}) - c^{2} a^{2}] v^{2} - [c^{2} (a^{2} + b^{2}) - a^{2} b^{2}] w^{2} - (a^{2} + b^{2}) u^{2} v^{2} - (b^{2} + c^{2}) v^{2} w^{2} - (c^{2} + a^{2}) w^{2} u^{2} + u^{2} v^{2} w^{2}$ $\Rightarrow {A R}^{2} + 2 B R - C = 0$ $\Rightarrow R = \frac{- B \pm \sqrt{B^{2} + A C}}{A}$ $e x a m p l e :$ $u = 7, v = 6, w = 8$ $a = 3, b = 2, c = 1$ $A = - (7^{4} + 6^{4} + 8^{4}) + 2 (7^{2} 6^{2} + 6^{2} 8^{2} + 8^{2} 7^{2}) + 4 [3 (2 + 1) - 2 \times 1 - 3^{2}] 7^{2} + 4 [2 (1 + 3) - 1 \times 3 - 2^{2}] 6^{2} + 4 [1 (3 + 2) - 3 \times 2 - 1^{2}] 8^{2} = 5855$ $B = 3 \times 7^{4} + 2 \times 6^{4} + 1 \times 8^{4} + 2 (3^{3} 7^{2} + 2^{3} 6^{2} + 1^{3} 8^{2}) - (3 + 2) (7^{2} 6^{2} + 1^{2} 8^{2}) - (2 + 1) (6^{2} 8^{2} + 3^{2} 7^{2}) - (1 + 3) (8^{2} 7^{2} + 2^{2} 6^{2}) + [2 \times 1 (2 + 1) - 3 (2^{2} + 1^{2})] 7^{2} + [1 \times 3 (1 + 3) - 2 (1^{2} + 3^{2})] 6^{2} + [3 \times 2 (3 + 2) - 1 (3^{2} + 2^{2})] 8^{2} = - 12895$ $C = 3^{2} 7^{2} (3^{2} + 7^{2}) + 2^{2} 6^{2} (2^{2} + 6^{2}) + 1^{2} 8^{2} (1^{2} + 8^{2}) - [3^{2} (2^{2} + 1^{2}) - 2^{2} 1^{2}] 7^{2} - [2^{2} (1^{2} + 3^{2}) - 1^{2} 3^{2}] 6^{2} - [1^{2} (3^{2} + 2^{2}) - 3^{2} 2^{2}] 8^{2} - (3^{2} + 2^{2}) 7^{2} 6^{2} - (2^{2} + 1^{2}) 6^{2} 8^{2} - (1^{2} + 3^{2}) 8^{2} 7^{2} + 7^{2} 6^{2} 8^{2} = 80929$ $R = \frac{12895 \pm \sqrt{12895^{2} + 5855 \times 80929}}{5855}$ $= \frac{2579}{1171} \pm \frac{3024 \sqrt{70}}{5855}$ $\approx 6.523586 / - 2.118804$
	Commented by mr W last updated on 19/May/24

	Answered by ajfour last updated on 18/May/24

	$s a y A i s o r i g i n . A O y a x i s .$ $r e d c i r c l e x^{2} + {(y - R + a)}^{2} = R^{2}$ $x_{B}^{2} + y_{B}^{2} = w^{2}$ $x_{C}^{2} + y_{C}^{2} = v^{2}$ $x_{B}^{2} + {(y_{B} - R + a)}^{2} = {(R - b)}^{2}$ $x_{C}^{2} + {(y_{C} - R + a)}^{2} = {(R - c)}^{2}$ $\Rightarrow w^{2} - 2 (R - a) y_{B} = {(R - b)}^{2}$ $& v^{2} - 2 (R - a) y_{c} = {(R - c)}^{2}$ $x_{B} = \sqrt{w^{2} - y_{B}^{2}}$ $x_{C} = - \sqrt{v^{2} - y_{C}^{2}}$ ${(x_{B} - x_{C})}^{2} + {(y_{B} - y_{C})}^{2} = u^{2}$ $★$
	Commented by mr W last updated on 18/May/24

	$w e l c o m e b a c k s i r!$
	Commented by ajfour last updated on 18/May/24

	$t h a n k s, b u t c o u l d u f o l l o w m y s o l u t i o n$ $s i r . .$
	Commented by mr W last updated on 19/May/24

	$y o u r p a t h i s r i g h t, t h a n k s s i r!$
	Commented by ajfour last updated on 18/May/24

	$c a n t h i s d o ?$ $Σ \cos^{- 1} \frac{{(R - b)}^{2} + {(R - c)}^{2} - u^{2}}{2 (R - b) (R - c)} = 2 π$
	Commented by mr W last updated on 19/May/24

	$y e s, t h i s w o r k s e i t h e r . b u t t h i s$ $e q u a t i o n i s n o t p o l y n o m i a l .$ $t h e d i s t a n c e s f r o m a p o i n t t o t h e$ $v e r t i c e s o f a t r i a n g l e f u l f i l l s$ $a p o l y n o m i a l e q u a t i o n .$
	Commented by mr W last updated on 19/May/24

	$t h e f i n a l e q u a t i o n c a n b e s i m p l i e d t o$ $a q u a d r a t i c e q u a t i o n, s o w e c a n g e t$ $t h e e x a c t s o l u t i o n . s e e a b o v e .$
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