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Question Number 202925 by mr W last updated on 05/Jan/24

Commented by mr W last updated on 05/Jan/24

$[Q 201146 r e p o s t e d]$ $f i n d I J = ?$
Commented by MathematicalUser2357 last updated on 09/Jan/24

$q n u m 201146$
	Answered by mr W last updated on 06/Jan/24

	$p = \frac{a + b + c}{2} (s e m i - p e r i m e t e r)$ $r = r a d i u s o f i n c i r c l e$ $R = r a d i u s o f c i r c u m c i r c l e$ $S = a r e a o f Δ A B C$ $= p r = \frac{a b c}{4 R} = \sqrt{p (p - a) (p - b) (p - c)}$ $A E = A F = p - a$ $B D = B F = p - b$ $C D = C E = p - c$ ${A I}^{2} = {(p - a)}^{2} + r^{2}$ $i n Δ A B C :$ ${a A D}^{2} = (p - b) b^{2} + (p - c) c^{2} - a (p - b) (p - c)$ ${A D}^{2} = \frac{(p - b) b^{2} + (p - c) c^{2}}{a} - (p - b) (p - c)$ ${A D}^{2} = \frac{(p - a) a^{2} + (p - b) b^{2} + (p - c) c^{2} - a b c}{a} + {(p - a)}^{2}$ ${A D}^{2} = \frac{p (a^{2} + b^{2} + c^{2}) - (a^{3} + b^{3} + c^{3}) - a b c}{a} + {(p - a)}^{2}$ ${A D}^{2} = \frac{p (2 p^{2} - 2 r^{2} - 8 R r) - (2 p^{3} - 6 r^{2} p - 12 R r p) - a b c}{a} + {(p - a)}^{2}$ ${A D}^{2} = \frac{4 r^{2} p + 4 R r p - a b c}{a} + {(p - a)}^{2}$ ${A D}^{2} = \frac{4 {p r}^{2} + 4 R S - 4 R S}{a} + {(p - a)}^{2}$ ${A D}^{2} = \frac{4 {p r}^{2}}{a} + {(p - a)}^{2}$ $\Rightarrow A D = \sqrt{\frac{4 {p r}^{2}}{a} + {(p - a)}^{2}}$ $\frac{A J}{J D} \times \frac{B D}{B C} \times \frac{C E}{E A} = 1$ $\frac{A J}{J D} \times \frac{p - b}{a} \times \frac{p - c}{p - a} = 1$ $\Rightarrow \frac{A J}{J D} = \frac{a (p - a)}{(p - b) (p - c)}$ $\Rightarrow A J = \frac{a (p - a)}{a (p - a) + (p - b) (p - c)} \times A D$ $= \frac{4 a (p - a)}{2 (a b + b c + c a) - (a^{2} + b^{2} + c^{2})} \times A D$ $= \frac{4 a (p - a)}{2 (p^{2} + r^{2} + 4 R r) - (2 p^{2} - 2 r^{2} - 8 R r)} \times A D$ $= \frac{a (p - a)}{r^{2} + 4 R r} \times A D$ $\Rightarrow J D = \frac{(p - b) (p - c)}{a (p - a) + (p - b) (p - c)} \times A D$ $= \frac{(p - b) (p - c)}{r^{2} + 4 R r} \times A D$ $i n Δ I A D :$ $A D \times {I J}^{2} = A J \times {D I}^{2} + J D \times {A I}^{2} - A D \times A J \times J D$ ${I J}^{2} = \frac{A J}{A D} \times {D I}^{2} + \frac{J D}{A D} \times {A I}^{2} - A J \times J D$ ${I J}^{2} = \frac{a (p - a)}{r^{2} + 4 R r} \times r^{2} + \frac{(p - b) (p - c)}{r^{2} + 4 R r} \times [r^{2} + {(p - a)}^{2}] - \frac{a (p - a)}{r^{2} + 4 R r} \times \frac{(p - b) (p - c)}{r^{2} + 4 R r} \times {A D}^{2}$ ${I J}^{2} = \frac{a (p - a) + (p - b) (p - c)}{r^{2} + 4 R r} \times r^{2} + \frac{{(p - a)}^{2} (p - b) (p - c)}{r^{2} + 4 R r} - \frac{a (p - a) (p - b) (p - c)}{{(r^{2} + 4 R r)}^{2}} \times {A D}^{2}$ ${I J}^{2} = \frac{r^{2} + 4 R r}{r^{2} + 4 R r} \times r^{2} + \frac{(p - a) S^{2}}{(r^{2} + 4 R r) p} - \frac{{a S}^{2}}{{(r^{2} + 4 R r)}^{2} p} \times {A D}^{2}$ ${I J}^{2} = r^{2} + \frac{{p r}^{2}}{{(r^{2} + 4 R r)}^{2}} [(p - a) (r^{2} + 4 R r) - a \times {A D}^{2}]$ ${I J}^{2} = r^{2} + \frac{p}{{(r + 4 R)}^{2}} [(p - a) (r^{2} + 4 R r) - 4 {p r}^{2} - a {(p - a)}^{2}]$ ${I J}^{2} = r^{2} + \frac{p}{{(r + 4 R)}^{2}} [(p - a) (r^{2} + 4 R r - a p + a^{2}) - 4 {p r}^{2}]$ ${I J}^{2} = r^{2} + \frac{p}{{(r + 4 R)}^{2}} [(p - a) (b c - (p - a) p) - 4 {p r}^{2}]$ ${I J}^{2} = r^{2} + \frac{p^{2}}{4 {(r + 4 R)}^{2}} [2 (a b + b c + c a) - (a^{2} + b^{2} + c^{2}) - 16 R r - 16 r^{2}]$ ${I J}^{2} = r^{2} + \frac{p^{2}}{4 {(r + 4 R)}^{2}} [2 p^{2} + 2 r^{2} + 8 R r - 2 p^{2} + 2 r^{2} + 8 R r - 16 R r - 16 r^{2}]$ ${I J}^{2} = r^{2} - \frac{3 p^{2} r^{2}}{{(r + 4 R)}^{2}}$ $\Rightarrow I J = r \sqrt{1 - \frac{3 p^{2}}{{(r + 4 R)}^{2}}}$
	Commented by mr W last updated on 06/Jan/24

	$y e s, y o u d i d . i {d i d n}^{'} t e x p e c t t h a t t h e$ $f o r m u l a c a n b e s o c o m p a c t .$
	Commented by ajfour last updated on 06/Jan/24

	$I r e m e m b e r i h a d t h e h o p e n$ $i n t u i t i o n t h a t s o m e t h i n g c o m p a c t$ $n b e a u t i f u l d h o u l d r e s u l t . T h a n k s!$
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