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Question Number 161912 by mr W last updated on 24/Dec/21

Commented by mr W last updated on 24/Dec/21

Commented by mr W last updated on 24/Dec/21

$i t i s t o f i n d a t r i a n g l e w h o s e s i d e s$ $t a n g e n t t h e t h r e e c i r c l e s w i t h r a d i i$ $p, q, r r e s p e c t i v e l y .$
Commented by mr W last updated on 24/Dec/21

$i f t h e d i s t a n c e s f r o m t h e o r t h o c e n t e r$ $o f a t r i a n g l e t o i t s s i d e s a r e p, q, r$ $r e s p e c t i v e l y, f i n d t h e s i d e l e n g t h e s$ $o f t h e t r i a n g l e .$
Commented by henderson last updated on 24/Dec/21

$very interesting . . .$ $i want to know it!$
Commented by mr W last updated on 25/Dec/21

Commented by mr W last updated on 25/Dec/21

$g e n e r a l l y w e c a n f i n d t w o t r i a n g l e s$ $u n d e r t h e g i v e n c o n d i t i o n .$ $r e d t r i a n g l e w i t h o r t h o c e n t e r i n s i d e$ $b l u e t r i a n g l e w i t h o r t h o c e n t e r o u t s i d e$
	Answered by mr W last updated on 25/Dec/21

	Commented by mr W last updated on 26/Dec/21
	$(p/v)=(q/u) ⇒pu=qv (p/w)=(r/u) ⇒pu=rw ⇒ determinant (((pu=qv=rw)))=(1/k), say ⇒u=(1/(pk)), v=(1/(qk)), w=(1/(rk)) say the area of triangle ABC is Δ and the side lengthes are a, b, c. (1/2)a(p+u)=(1/2)b(q+v)=(1/2)c(r+w)=Δ (1/2)(au+bv+cw)=2Δ (1/2)(((2Δu)/(p+u))+((2Δv)/(q+v))+((2Δw)/(r+w)))=2Δ ⇒ determinant ((((u/(p+u))+(v/(q+v))+(w/(r+w))=2))) (1/((p/u)+1))+(1/((q/v)+1))+(1/((r/w)+1))=2 (1/(p^2 k+1))+(1/(q^2 k+1))+(1/(r^2 k+1))=2 (p^2 k+1)(q^2 k+1)+(q^2 k+1)(r^2 k+1)+(r^2 k+1)(p^2 k+1)=2(p^2 k+1)(q^2 k+1)(r^2 k+1) (p^2 q^2 +q^2 r^2 +r^2 p^2 )k^2 +2(p^2 +q^2 +r^2 )k+3=2p^2 q^2 r^2 k^3 +2(p^2 q^2 +q^2 r^2 +r^2 p^2 )k^2 +2(p^2 +q^2 +r^2 )k+2 2p^2 q^2 r^2 k^3 +(p^2 q^2 +q^2 r^2 +r^2 p^2 )k^2 −1=0 determinant ((((1/k^3 )−(p^2 q^2 +q^2 r^2 +r^2 p^2 ) (1/k)−2(pqr)^2 =0))) (pqr)^4 −(((p^2 q^2 +q^2 r^2 +r^2 p^2 )/3))^3 ≤0 ⇒three real roots (1/k_n )=2(√((p^2 q^2 +q^2 r^2 +r^2 p^2 )/3)) sin {((2nπ)/3)−(1/3) sin^(−1) [(pqr)^2 ((3/(p^2 q^2 +q^2 r^2 +r^2 p^2 )))^(3/2) ]} (n=0,1,2) generally two (i.e. n=1,2) of the three roots are suitable: determinant ((((1/k_1 )=2(√((p^2 q^2 +q^2 r^2 +r^2 p^2 )/3)) sin {(π/3)+(1/3) sin^(−1) [(pqr)^2 ((3/(p^2 q^2 +q^2 r^2 +r^2 p^2 )))^(3/2) ]})),(((1/k_2 )=−2(√((p^2 q^2 +q^2 r^2 +r^2 p^2 )/3)) sin {(π/3)−(1/3) sin^(−1) [(pqr)^2 ((3/(p^2 q^2 +q^2 r^2 +r^2 p^2 )))^(3/2) ]}))) the positive value is for the case that the orthocenter lies inside the triangle and the negative value for the case that it lies outside the triangle. with k we get 𝚫=(1/( (√(((1/(p+(1/(pk))))+(1/(q+(1/(qk))))+(1/(r+(1/(rk)))))(−(1/(p+(1/(pk))))+(1/(q+(1/(qk))))+(1/(r+(1/(rk)))))((1/(p+(1/(pk))))−(1/(q+(1/(qk))))+(1/(r+(1/(rk)))))((1/(p+(1/(pk))))+(1/(q+(1/(qk))))−(1/(r+(1/(rk))))))))) a=((2𝚫)/(p+(1/(pk)))), b=((2𝚫)/(q+(1/(qk)))), c=((2𝚫)/(r+(1/(rk)))) example: p=10, q=6, r=4 a≈28.522248 or 15.173152 b≈26.202978 or 12.655577 c≈20.945175 or 4.829686$
	$\frac{p}{v} = \frac{q}{u} \Rightarrow p u = q v$ $\frac{p}{w} = \frac{r}{u} \Rightarrow p u = r w$ $\Rightarrow \begin{matrix} p u = q v = r w \end{matrix} = \frac{1}{k}, s a y$ $\Rightarrow u = \frac{1}{p k}, v = \frac{1}{q k}, w = \frac{1}{r k}$ $s a y t h e a r e a o f t r i a n g l e A B C i s Δ$ $a n d t h e s i d e l e n g t h e s a r e a, b, c .$ $\frac{1}{2} a (p + u) = \frac{1}{2} b (q + v) = \frac{1}{2} c (r + w) = Δ$ $\frac{1}{2} (a u + b v + c w) = 2 Δ$ $\frac{1}{2} (\frac{2 Δ u}{p + u} + \frac{2 Δ v}{q + v} + \frac{2 Δ w}{r + w}) = 2 Δ$ $\Rightarrow \begin{matrix} \frac{u}{p + u} + \frac{v}{q + v} + \frac{w}{r + w} = 2 \end{matrix}$ $\frac{1}{\frac{p}{u} + 1} + \frac{1}{\frac{q}{v} + 1} + \frac{1}{\frac{r}{w} + 1} = 2$ $\frac{1}{p^{2} k + 1} + \frac{1}{q^{2} k + 1} + \frac{1}{r^{2} k + 1} = 2$ $(p^{2} k + 1) (q^{2} k + 1) + (q^{2} k + 1) (r^{2} k + 1) + (r^{2} k + 1) (p^{2} k + 1) = 2 (p^{2} k + 1) (q^{2} k + 1) (r^{2} k + 1)$ $(p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}) k^{2} + 2 (p^{2} + q^{2} + r^{2}) k + 3 = 2 p^{2} q^{2} r^{2} k^{3} + 2 (p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}) k^{2} + 2 (p^{2} + q^{2} + r^{2}) k + 2$ $2 p^{2} q^{2} r^{2} k^{3} + (p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}) k^{2} - 1 = 0$ $\begin{matrix} \frac{1}{k^{3}} - (p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}) \frac{1}{k} - 2 {(p q r)}^{2} = 0 \end{matrix}$ ${(p q r)}^{4} - {(\frac{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}}{3})}^{3} ⩽ 0 \Rightarrow t h r e e r e a l r o o t s$ $\frac{1}{k_{n}} = 2 \sqrt{\frac{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}}{3}} \sin {\frac{2 n π}{3} - \frac{1}{3} \sin^{- 1} [{(p q r)}^{2} {(\frac{3}{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}})}^{\frac{3}{2}}]} (n = 0, 1, 2)$ $g e n e r a l l y t w o (i . e . n = 1, 2) o f t h e t h r e e$ $r o o t s a r e s u i t a b l e :$ $\begin{array}{cc} \frac{1}{k_{1}} = 2 \sqrt{\frac{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}}{3}} \sin {\frac{π}{3} + \frac{1}{3} \sin^{- 1} [{(p q r)}^{2} {(\frac{3}{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}})}^{\frac{3}{2}}]} \\ \frac{1}{k_{2}} = - 2 \sqrt{\frac{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}}{3}} \sin {\frac{π}{3} - \frac{1}{3} \sin^{- 1} [{(p q r)}^{2} {(\frac{3}{p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}})}^{\frac{3}{2}}]} \end{array}$ $t h e p o s i t i v e v a l u e i s f o r t h e c a s e t h a t$ $t h e o r t h o c e n t e r l i e s i n s i d e t h e t r i a n g l e$ $a n d t h e n e g a t i v e v a l u e f o r t h e c a s e$ $t h a t i t l i e s o u t s i d e t h e t r i a n g l e .$ $w i t h k w e g e t$ $Δ = \frac{1}{\sqrt{(\frac{1}{p + \frac{1}{p k}} + \frac{1}{q + \frac{1}{q k}} + \frac{1}{r + \frac{1}{r k}}) (- \frac{1}{p + \frac{1}{p k}} + \frac{1}{q + \frac{1}{q k}} + \frac{1}{r + \frac{1}{r k}}) (\frac{1}{p + \frac{1}{p k}} - \frac{1}{q + \frac{1}{q k}} + \frac{1}{r + \frac{1}{r k}}) (\frac{1}{p + \frac{1}{p k}} + \frac{1}{q + \frac{1}{q k}} - \frac{1}{r + \frac{1}{r k}})}}$ $a = \frac{2 Δ}{p + \frac{1}{p k}}, b = \frac{2 Δ}{q + \frac{1}{q k}}, c = \frac{2 Δ}{r + \frac{1}{r k}}$ $\underset{―}{e x a m p l e : p = 10, q = 6, r = 4}$ $a \approx 28.522248 o r 15.173152$ $b \approx 26.202978 o r 12.655577$ $c \approx 20.945175 o r 4.829686$
	Commented by mr W last updated on 25/Dec/21

	Commented by mr W last updated on 25/Dec/21

	Commented by mr W last updated on 25/Dec/21

	Commented by Ar Brandon last updated on 25/Dec/21

	${What}^{'} s all this, Sir$ 🙁☹️
	Commented by mr W last updated on 25/Dec/21

	$h o w t o f i n d t h e t r i a n g l e i f y o u o n l y$ $k n o w t h e d i s t a n c e s f r o m i t s$ $o r t h o c e n t e r t o i t s t h r e e s i d e s .$
	Commented by Ar Brandon last updated on 25/Dec/21

	😭
	Commented by Tawa11 last updated on 25/Dec/21

	$Wow, great sir .$
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