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Question Number 149428 by mr W last updated on 06/Aug/21

Commented by mr W last updated on 06/Aug/21

$t h r e e n o n - c o l l i n e a r p o i n t s i n s p a c e$ $a r e g i v e n w i t h c o o r d i n a t e s$ $A (x_{1}, y_{1}, z_{1})$ $B (x_{2}, y_{2}, z_{2})$ $C (x_{3}, y_{3}, z_{3})$ $f i n d t h e c o o r d i n a t e s o f t h e c e n t e r S$ $a n d t h e r a d i u s R o f t h e c i r c u m c i r c l e .$
Commented by ajfour last updated on 06/Aug/21
$sir, (apart from this); sir try plotting z(x,y)={(x^2 −1)^3 +(y^2 −1)^3 }^2 see all views, change scale, wonderful (i find)...$
$s i r, (a p a r t f r o m t h i s); s i r t r y p l o t t i n g$ $z (x, y) = {{(x^{2} - 1)}^{3} + {(y^{2} - 1)}^{3}}^{2}$ $s e e a l l v i e w s, c h a n g e s c a l e,$ $w o n d e r f u l (i f i n d) . . .$
Commented by ajfour last updated on 06/Aug/21

	Answered by ajfour last updated on 06/Aug/21
	$A(a), B(b), C(c) plane ABC (s−a).((b−c)×(a−c))=0 let S(s). n^� =(((b−c)×(a−c))/(∣(b−c)×(a−c)∣)) let s=(((b+c))/2)+((∣b−c∣)/(2tan A)){((n^� ×(b−c))/( ∣b−c∣))} ∣s−a∣=∣s−b∣=∣s−c∣=R$
	$A (a), B (b), C (c)$ $p l a n e A B C$ $(s - a) . ((b - c) \times (a - c)) = 0$ $l e t S (s) .$ $\hat{n} = \frac{(b - c) \times (a - c)}{∣ (b - c) \times (a - c) ∣}$ $l e t s = \frac{(b + c)}{2} + \frac{∣ b - c ∣}{2 \tan A} {\frac{\hat{n} \times (b - c)}{∣ b - c ∣}}$ $∣ s - a ∣=∣ s - b ∣=∣ s - c ∣= R$
	Commented by mr W last updated on 06/Aug/21

	$t h a n k s s i r! i^{'} l l a l s o t r y t o u s e v e c t o r s .$
	Commented by ajfour last updated on 06/Aug/21

	$l e t A (5, 2, 3); B (3, 5, 5);$ $C (2, 3, 8)$ $\hat{n} = \frac{(i + 2 j - 3 k) \times (2 i - 3 j - 2 k)}{∣ (i + 2 j - 3 k) \times (2 i - 3 j - 2 k) ∣}$ $= \frac{- 3 k + 2 j - 4 k + 4 i + 6 j - 9 i}{\sqrt{25 + 64 + 49}}$ $= \frac{- 5 i + 8 j - 7 k}{\sqrt{138}}$ $s = \frac{(5 i + 8 j + 13 k)}{2}$ $+ \frac{∣ i + 2 j - 3 k ∣}{2 \tan A} (\frac{- 5 i + 8 j - 7 k}{\sqrt{138}} \times \frac{i + 2 j - 3 k}{\sqrt{14}})$ $l e t D b e m i d - p o i n t o f A B = \frac{a + b}{2}$ $C D = \frac{a + b - 2 c}{2} = \frac{4 i + j - 8 k}{2}$ $\frac{A B}{2} = \frac{- 2 i + 3 j + 2 k}{2}$ $\tan A = \frac{C D}{(A B) / 2} = \frac{\sqrt{16 + 1 + 64}}{\sqrt{4 + 9 + 4}}$ $= \frac{9}{\sqrt{17}}$ $s = \frac{5 i + 8 j + 13 k}{2}$ $+ \frac{\sqrt{17}}{18 \sqrt{138}} (- 10 k - 15 j - 8 k - 24 i$ $- 7 j + 14 i)$ $s = \frac{5 i + 8 j + 13 k}{2}$ $+ \frac{\sqrt{17} (- 10 i - 22 j - 18 k)}{18 \sqrt{138}}$ $R =∣ s - a ∣$
	Commented by mr W last updated on 07/Aug/21

	$i g o t i t! {i t}^{'} s a g r e a t i d e a w i t h \frac{1}{\tan A}!$
	Commented by ajfour last updated on 07/Aug/21

	Answered by mr W last updated on 07/Aug/21

	Commented by mr W last updated on 07/Aug/21
	$let a=OA^(→) , b=OB^(→) , c=OC^(→) , s=OS^(→) let p=AB^(→) , q=BC^(→) , r=CA^(→) normal of plane ABC: n=((r×q)/(∣r×q∣)) D=midpoint of AB OD^(→) =((a+b)/2) ∣DS^(→) ∣=((∣AB^(→) ∣)/2)tan φ=((∣p∣)/(2 tan ∠C)) DS^(→) =∣DS^(→) ∣((p×n)/(∣p∣))=((∣p∣)/(2 tan ∠C))×((p×(r×q))/(∣p∣∣r×q∣)) DS^(→) =(1/(tan ∠C))×((p×(r×q))/(2∣r×q∣)) s=OD^(→) +DS^(→) s=((a+b)/2)+(1/(tan ∠C))×((p×(r×q))/(2∣r×q∣)) cos C=−((q∙r)/(∣q∣∣r∣)) sin C=((∣q×r∣)/(∣q∣∣r∣)) (1/(tan C))=−((q∙r)/(∣q×r∣)) s=((a+b)/2)−((q∙r)/(∣q×r∣))×((p×(r×q))/(2∣r×q∣)) ⇒s=(1/2){a+b+((q∙r)/(∣q×r∣^2 )) p×(q×r)} ⇒R=∣s−a∣=∣s−b∣=∣s−c∣ or R=((∣p∣)/(2 sin ∠C))=((∣p∣∣q∣∣r∣)/(2∣q×r∣)) example: A(5,2,3), B(3,5,5), C(2,3,8) p=(−2,3,2) q=(−1,−2,3) r=(3,−1,−5) q∙r=−3+2−15=−16 q×r= determinant (((−1),(−2),3),(3,(−1),(−5)))=(13,4,7) ∣q×r∣^2 =13^2 +4^2 +7^2 =234 p×(q×r)= determinant (((−2),3,2),((13),4,7))=(13,40,−47) s=(1/2){(5,2,3)+(3,5,5)−((16)/(234))(13,40,−47)} =(((32)/9),((499)/(234)),((656)/(117))) R^2 =∣AS^(→) ∣^2 =(((32)/9)−5)^2 +(((499)/(234))−2)^2 +(((656)/(117))−3)^2 =((4165)/(468)) ✓ R^2 =∣BS^(→) ∣^2 =(((32)/9)−3)^2 +(((499)/(234))−5)^2 +(((656)/(117))−5)^2 =((4165)/(468)) ✓ R^2 =∣CS^(→) ∣^2 =(((32)/9)−2)^2 +(((499)/(234))−3)^2 +(((656)/(117))−8)^2 =((4165)/(468)) ✓ R=(√((4165)/(468)))=((7(√(1105)))/(78))$
	$l e t a = \vec{O A}, b = \vec{O B}, c = \vec{O C}, s = \vec{O S}$ $l e t p = \vec{A B}, q = \vec{B C}, r = \vec{C A}$ $n o r m a l o f p l a n e A B C :$ $n = \frac{r \times q}{∣ r \times q ∣}$ $D = m i d p o i n t o f A B$ $\vec{O D} = \frac{a + b}{2}$ $∣ \vec{D S} ∣= \frac{∣ \vec{A B} ∣}{2} \tan ϕ = \frac{∣ p ∣}{2 \tan ∠ C}$ $\vec{D S} =∣ \vec{D S} ∣ \frac{p \times n}{∣ p ∣} = \frac{∣ p ∣}{2 \tan ∠ C} \times \frac{p \times (r \times q)}{∣ p ∣∣ r \times q ∣}$ $\vec{D S} = \frac{1}{\tan ∠ C} \times \frac{p \times (r \times q)}{2 ∣ r \times q ∣}$ $s = \vec{O D} + \vec{D S}$ $s = \frac{a + b}{2} + \frac{1}{\tan ∠ C} \times \frac{p \times (r \times q)}{2 ∣ r \times q ∣}$ $\cos C = - \frac{q \cdot r}{∣ q ∣∣ r ∣}$ $\sin C = \frac{∣ q \times r ∣}{∣ q ∣∣ r ∣}$ $\frac{1}{\tan C} = - \frac{q \cdot r}{∣ q \times r ∣}$ $s = \frac{a + b}{2} - \frac{q \cdot r}{∣ q \times r ∣} \times \frac{p \times (r \times q)}{2 ∣ r \times q ∣}$ $\Rightarrow s = \frac{1}{2} {a + b + \frac{q \cdot r}{∣ q \times r ∣^{2}} p \times (q \times r)}$ $\Rightarrow R =∣ s - a ∣=∣ s - b ∣=∣ s - c ∣$ $o r R = \frac{∣ p ∣}{2 \sin ∠ C} = \frac{∣ p ∣∣ q ∣∣ r ∣}{2 ∣ q \times r ∣}$ $e x a m p l e :$ $A (5, 2, 3), B (3, 5, 5), C (2, 3, 8)$ $p = (- 2, 3, 2)$ $q = (- 1, - 2, 3)$ $r = (3, - 1, - 5)$ $q \cdot r = - 3 + 2 - 15 = - 16$ $q \times r = \| \begin{matrix} - 1 & - 2 & 3 \\ 3 & - 1 & - 5 \end{matrix} \| = (13, 4, 7)$ $∣ q \times r ∣^{2} = 13^{2} + 4^{2} + 7^{2} = 234$ $p \times (q \times r) = \| \begin{matrix} - 2 & 3 & 2 \\ 13 & 4 & 7 \end{matrix} \| = (13, 40, - 47)$ $s = \frac{1}{2} {(5, 2, 3) + (3, 5, 5) - \frac{16}{234} (13, 40, - 47)}$ $= (\frac{32}{9}, \frac{499}{234}, \frac{656}{117})$ $R^{2} =∣ \vec{A S} ∣^{2} = {(\frac{32}{9} - 5)}^{2} + {(\frac{499}{234} - 2)}^{2} + {(\frac{656}{117} - 3)}^{2} = \frac{4165}{468} ✓$ $R^{2} =∣ \vec{B S} ∣^{2} = {(\frac{32}{9} - 3)}^{2} + {(\frac{499}{234} - 5)}^{2} + {(\frac{656}{117} - 5)}^{2} = \frac{4165}{468} ✓$ $R^{2} =∣ \vec{C S} ∣^{2} = {(\frac{32}{9} - 2)}^{2} + {(\frac{499}{234} - 3)}^{2} + {(\frac{656}{117} - 8)}^{2} = \frac{4165}{468} ✓$ $R = \sqrt{\frac{4165}{468}} = \frac{7 \sqrt{1105}}{78}$
	Commented by mr W last updated on 07/Aug/21
	$a special case A(u,0,0) B(0,v,0) C(0,0,w) p=(−u,v,0) q=(0,−v,w) r=(u,0,−w) q∙r=−w^2 q×r= determinant ((0,(−v),w),(u,0,(−w)))=(vw,wu,uv) ∣q×r∣^2 =u^2 v^2 +v^2 w^2 +w^2 u^2 p×(q×r)= determinant (((−u),v,0),((vw),(wu),(uv)))=(uv^2 ,u^2 v,−w(u^2 +v^2 )) s=(1/2){(u,0,0)+(0,v,0)−(w^2 /(u^2 v^2 +v^2 w^2 +w^2 u^2 )) (uv^2 ,u^2 v,−w(u^2 +v^2 ))} =(1/2){u−((uv^2 w^2 )/(u^2 v^2 +v^2 w^2 +w^2 u^2 )),v−((u^2 vw^2 )/(u^2 v^2 +v^2 w^2 +w^2 u^2 )),((w^3 (u^2 +v^2 ))/(u^2 v^2 +v^2 w^2 +w^2 u^2 ))} =(1/2){((u^3 (v^2 +w^2 ))/(u^2 v^2 +v^2 w^2 +w^2 u^2 )),((v^3 (w^2 +u^2 ))/(u^2 v^2 +v^2 w^2 +w^2 u^2 )),((w^3 (u^2 +v^2 ))/(u^2 v^2 +v^2 w^2 +w^2 u^2 ))} ∣p∣=(√(u^2 +v^2 )) ∣q∣=(√(v^2 +w^2 )) ∣r∣=(√(w^2 +u^2 )) ∣q×r∣=(√(u^2 v^2 +v^2 w^2 +w^2 u^2 )) ⇒R=(1/2)(√(((u^2 +v^2 )(v^2 +w^2 )(w^2 +u^2 ))/(u^2 v^2 +v^2 w^2 +w^2 u^2 )))$
	$a s p e c i a l c a s e$ $A (u, 0, 0)$ $B (0, v, 0)$ $C (0, 0, w)$ $p = (- u, v, 0)$ $q = (0, - v, w)$ $r = (u, 0, - w)$ $q \cdot r = - w^{2}$ $q \times r = \| \begin{matrix} 0 & - v & w \\ u & 0 & - w \end{matrix} \| = (v w, w u, u v)$ $∣ q \times r ∣^{2} = u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}$ $p \times (q \times r) = \| \begin{matrix} - u & v & 0 \\ v w & w u & u v \end{matrix} \| = ({u v}^{2}, u^{2} v, - w (u^{2} + v^{2}))$ $s = \frac{1}{2} {(u, 0, 0) + (0, v, 0) - \frac{w^{2}}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}} ({u v}^{2}, u^{2} v, - w (u^{2} + v^{2}))}$ $= \frac{1}{2} {u - \frac{{u v}^{2} w^{2}}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}, v - \frac{u^{2} {v w}^{2}}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}, \frac{w^{3} (u^{2} + v^{2})}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}}$ $= \frac{1}{2} {\frac{u^{3} (v^{2} + w^{2})}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}, \frac{v^{3} (w^{2} + u^{2})}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}, \frac{w^{3} (u^{2} + v^{2})}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}}$ $∣ p ∣= \sqrt{u^{2} + v^{2}}$ $∣ q ∣= \sqrt{v^{2} + w^{2}}$ $∣ r ∣= \sqrt{w^{2} + u^{2}}$ $∣ q \times r ∣= \sqrt{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}$ $\Rightarrow R = \frac{1}{2} \sqrt{\frac{(u^{2} + v^{2}) (v^{2} + w^{2}) (w^{2} + u^{2})}{u^{2} v^{2} + v^{2} w^{2} + w^{2} u^{2}}}$
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