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Question Number 39300 by behi83417@gmail.com last updated on 05/Jul/18

	Answered by MrW3 last updated on 05/Jul/18

	$a s q u a r e c a n i n s c r i b e e v e r y t r i a n g l e .$ $s o t h e q u e s t i o n i s$ $i f i t i s p o s s i b l e t o b u i l d a t r i a n g l e$ $w h o s e s i d e s a r e t h r e e c o n s e c u t i v e o d d$ $(o r e v e n) n u m b e r s .$ $a = n$ $b = n + 2$ $c = n + 4$ $f o r t h i s t o b e p o s s i b l e, w e m u s t h a v e$ $a + b > c \Rightarrow 2 n + 2 > n + 4 \Rightarrow n > 2$ $b + c > a \Rightarrow 2 n + 6 > n \Rightarrow n > - 6$ $c + a > b \Rightarrow 2 n + 4 > n + 2 \Rightarrow n > - 2$ $t h a t m e a n s n ⩾ 3 .$ $s o t h e a n s w e r i s y e s .$ $i f a d d i t i o n a l l y t h e t r i a n g l e s h o u l d o n l y$ $h a v e a c u t e a n g l e s, w e m u s t h a v e$ ${(n + 4)}^{2} ⩽ n^{2} + {(n + 2)}^{2}$ $n^{2} + 8 n + 16 ⩽ n^{2} + n^{2} + 4 n + 4$ $n^{2} - 4 n - 12 ⩾ 0$ $(n + 2) (n - 6) ⩾ 0$ $n ⩾ 6$
	Commented by behi83417@gmail.com last updated on 05/Jul/18

	$d e a r m a s t e r! t h a n k s a l o t . I g o t t h e$ $a n s w e r .$ $c a n w e p r o v e t h e f i r s t s t a t e m e n t ?$
	Commented by MrW3 last updated on 05/Jul/18

	$S t a t e m e n t :$ $I n e v e r y t r i a n g l e i t i s p o s s i b l e t o$ $f i n d a n i n s c r i b i n g s q u a r e .$ $P r o o f :$ $I n e v e r y t r i a n g l e t h e r e a r e a t l e a s t$ $t w o a c u t e a n g l e s . A s s u m e t h e s i d e$ $l e n g t h b e t w e e n t h e s e t w o a c u t e a n g l e s$ $i s a, a n d c o r r e s p o n d i n g a l t i t u d e i s h .$ $D E = \frac{x}{h} \times a$ $D F = h - x$ $D E = D F$ $\frac{x}{h} a = h - x$ $\Rightarrow x = \frac{h}{h + a} \times h$ $0 < x < h$ $i . e . t h e r e i s a x = \frac{h}{h + a} \times a w i t h w h i c h$ $D E G F i s a s q u a r e .$ $S i n c e ∠ B a n d ∠ C a r e a c u t e,$ $\Rightarrow F a n d G l i e o n t h e s i d e B C,$ $i . e . D E G F i s a n i n s c r i b i n g s q u a r e .$
	Commented by MrW3 last updated on 05/Jul/18

	Commented by behi83417@gmail.com last updated on 05/Jul/18

	$t h a n k s i n a d v a n c e d e a r m a s t e r .$ $n i c e p r o o f . I t h i n k i t i s t h e f i r s t p r o o f$ $t o t h i s s t a t e m e n t .$
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