Question-15927

Question Number 15927 by ajfour last updated on 15/Jun/17

Commented by ajfour last updated on 15/Jun/17

Q . 15917 (s o r r y i t g o t u p l o a d e d

a s n e w q u e s t i o n)

m i d p o i n t s o f t h e s i d e s o f

q u a d r i l a t e r a l A B C D a r e

E, F, G, a n d H . M i d p o i n t s o f

i t s d i g o n a l s L a n d K .

T o p r o v e : i n t e r s e c t i o n p o i n t J

o f E G a n d H F l i e s o n L K a n d

i s t h e i r (E G, H F, L K) m i d p o i n t .

Commented by ajfour last updated on 15/Jun/17

z_{1} + z_{2} = z_{3} + z_{4} \dots . . (i)

l e t B i s t h e o r i g i n

z_{L} = \frac{z_{1} + z_{2}}{2},

z_{K} = z_{1} + \frac{z_{3} - z_{1}}{2} = \frac{z_{1} + z_{3}}{2}

z_{H} = \frac{z_{1}}{2}, z_{F} = z_{3} + \frac{z_{4}}{2}

A p o i n t o f H F :

z_{H F} = z_{H} + μ (z_{F} - z_{H})

= \frac{z_{1}}{2} + μ (z_{3} + \frac{z_{4} - z_{1}}{2})

= \frac{z_{1}}{2} + μ (z_{3} + \frac{z_{2} - z_{3}}{2}) [s e e (i)]

z_{H F} = \frac{z_{1}}{2} + μ (\frac{z_{2} + z_{3}}{2}) \dots (i i)

z_{E} = \frac{z_{3}}{2}, z_{G} = z_{1} + \frac{z_{2}}{2}

A p o i n t o n E G o b e y s :

z_{E G} = z_{E} + ϵ (z_{G} - z_{E})

= \frac{z_{3}}{2} + ϵ (z_{1} + \frac{z_{2} - z_{3}}{2}) \dots (i i i)

p o i n t o f i n t e r s e c t i o n J o f H F a n d

E G s h a l l o b e y b o t h t h e i r e q n s .

(i i) a n d (i i i)

z_{J} = \frac{z_{1}}{2} + μ (\frac{z_{2} + z_{3}}{2}) = \frac{z_{3}}{2} + ϵ (z_{1} + \frac{z_{2} - z_{3}}{2})

\Rightarrow z_{1} (\frac{1}{2} - ϵ) + z_{2} (\frac{μ}{2} - \frac{ϵ}{2}) +

z_{3} (\frac{μ}{2} - \frac{1}{2} + \frac{ϵ}{2}) = 0

a s z_{1}, z_{2}, z_{3} c a n b e i n d e p e n d e n t l y

c h o s e n t o f o r m a q u a d r i l a t e r a l

b u t a b o v e e q n . i s a l w a y s t r u e, s o

f o r p o i n t J,

ϵ = \frac{1}{2}, μ = ϵ = \frac{1}{2}

t h i s i m p l i e s J i s t h e m i d p o i n t

o f H F a n d E G, a n d t h a t

z_{J} = \frac{z_{1}}{2} + \frac{z_{2} + z_{3}}{4}

(b y s u b s t i t u t i n g μ = \frac{1}{2} i n (i i))

I t n o w r e m a i n s t o p r o v e t h a t

J i s a l s o t h e m i d p o i n t o f L K .

m i d p o i n t o f L K i s

z = \frac{z_{L} + z_{K}}{2}

z = \frac{1}{2} (\frac{z_{1} + z_{2}}{2} + \frac{z_{1} + z_{3}}{2})

= \frac{z_{1}}{2} + \frac{z_{2} + z_{3}}{4} = z_{J}

s o m i d p o i n t s o f E G, H F, a n d

L K c o i n c i d e i n J .

Commented by mrW1 last updated on 15/Jun/17

excellent sir! you master the vector

technique outstandingly .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 19/Jun/17

l e t : A (a), B (b), C (c), D (d), L (l), K (k)

H (h) = \frac{b + c}{2}, F (f) = \frac{a + d}{2}, E (e) = \frac{a + b}{2},

G (g) = \frac{d + c}{2}

\Rightarrow L (l) = \frac{b + d}{2}, K (k) = \frac{a + c}{2}

J = \frac{\frac{b + d}{2} + \frac{a + c}{2}}{2} = \frac{a + b + c + d}{4} (i)

m i d p o i n t o f F H = \frac{\frac{a + d}{2} + \frac{b + c}{2}}{2} = \frac{a + d + b + c}{4} (i i)

m i d p o i n t o f E G = \frac{\frac{a + b}{2} + \frac{d + c}{2}}{2} = \frac{a + b + d + c}{4} (i i i)

a, b, c, d, e, f, g,, k, l, a r e c o m p l e x n u m b e r s .

f r o m e q u a t i o n s i, i i, i i i, w e s e e t h a t :

m i d p o i n t o f : E G, F H, L K, h a v e t h e s a m e

c o r d i n a t e s . s o t h e y s h o u l d b e t h e s a m e

p o i n t .

Commented by ajfour last updated on 19/Jun/17

t h a n k y o u, s i r . y o u c u t m y l o n g

s t o r y s h o r t .

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