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Question Number 52806 by mr W last updated on 13/Jan/19

Commented by mr W last updated on 13/Jan/19

$I f o u n d t h i s u n s o l v e d q u e s t i o n i n$ $Q 35115 (f r o m a j f o u r s i r) :$ $F i n d r i n t e r m s o f a, b, c .$
Commented by behi83417@gmail.com last updated on 14/Jan/19

	Answered by MJS last updated on 14/Jan/19

	$A = (\begin{matrix} 0 \\ 0 \end{matrix}) B = (\begin{matrix} c \\ 0 \end{matrix}) C = (\begin{matrix} b \cos α \\ b \sin α \end{matrix}) \tan \frac{α}{2} = k$ $circumcircle$ $center = (\begin{matrix} m \\ n \end{matrix}); radius = R = \sqrt{m^{2} + n^{2}}$ $blue circle$ $center = (\begin{matrix} o \\ k o \end{matrix}); radius = r = k o$ $(1) {(x - m)}^{2} + {(y - n)}^{2} - (m^{2} + n^{2}) = 0$ $x^{2} + y^{2} - 2 m x - 2 n y = 0$ $(2) {(x - o)}^{2} + {(y - k o)}^{2} - {(o k)}^{2} = 0$ $x^{2} + y^{2} - 2 o x - 2 k o y + o^{2} = 0$ $(1) - (2)$ $\Rightarrow y = \frac{2 (m - o) x + o^{2}}{2 (k o - n)}$ $insert in (1) or (2)$ $\Rightarrow x^{2} + u x + v = 0$ $we need exactly one solution \Rightarrow$ $\Rightarrow D = u^{2} - 4 v = 0$ $o^{4} - \frac{2 (2 k^{2} n + 2 k m + n)}{k} o^{3} - \frac{4 k^{4} m^{2} - 8 k^{3} m n - 4 k^{2} (m^{2} + 2 n^{2}) - 8 k m n - n^{2}}{k^{2}} o^{2} + \frac{4 (2 k^{3} m^{2} - 4 k^{2} m n - k (2 m^{2} + n^{2}) - m n) n}{k^{2}} o - \frac{4 (k^{2} m - 2 k n - m) {m n}^{2}}{k^{2}} = 0$ $trying the factors of the constant we get$ $o_{1} = o_{2} = \frac{n}{k} [not usable]$ $o_{3} = 2 k n + 2 m - 2 k \sqrt{m^{2} + n^{2}}$ $o_{4} = 2 k n + 2 m + 2 k \sqrt{m^{2} + n^{2}}$ $now insert$ $k = \frac{(a + b - c) (a - b + c)}{δ}$ $m = \frac{c}{2}$ $n = \frac{(a^{2} + b^{2} - c^{2}) c}{2 δ}$ $δ = \sqrt{(a + b + c) (a + b - c) (a - b + c) (- a + b + c)}$ $R = \frac{a b c}{δ}$ $o_{3} = \frac{2 b c}{a + b + c} o_{4} = \frac{2 b c}{- a + b + c}$ $r_{3} = \frac{2 (a + b - c) (a - b + c) b c}{(a + b + c) δ} [blue circle]$ $r_{4} = \frac{2 (a + b - c) (a - b + c) b c}{(- a + b + c) δ} [circle touching circumcircle from outside]$
	Commented by behi83417@gmail.com last updated on 15/Jan/19

	$t h a n k y o u s i r M J S . t h i s i s p e r f e c t .$ $c h e c k i n g : r_{3} \Rightarrow r i g h t a n s w e r .$
	Commented by MJS last updated on 15/Jan/19

	$thank you for checking Sir Behi$
	Commented by mr W last updated on 15/Jan/19

	$r = r_{3} = \frac{2 (a + b - c) (a - b + c) b c}{(a + b + c) δ}$ $r = \frac{2 b c (a + b - c) (a - b + c)}{(a + b + c) \sqrt{(a + b + c) (a + b - c) (a - b + c) (- a + b + c)}}$ $r = \frac{2 b c \sqrt{(a + b - c) (a - b + c)}}{(a + b + c) \sqrt{(a + b + c) (- a + b + c)}}$ $r = \frac{2 b c}{(a + b + c)} \sqrt{\frac{(a + b - c) (a + c - b)}{(a + b + c) (b + c - a)}}$ $t h i s i s c o r r e c t s i r .$
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