Let-ABCD-be-a-convex-quadrilateral-and-M-a-point-in-its-interior-such-that-MAB-MBC-MCD-MDA-Prove-that-one-of-the-diagonals-of-ABCD-passes-through-the-midpoint-of-the-other-diagonal-

Question Number 16064 by Tinkutara last updated on 21/Jun/17

Let A B C D be a convex quadrilateral

and M a point in its interior such that

[M A B] = [M B C] = [M C D] = [M D A] .

Prove that one of the diagonals of

A B C D passes through the midpoint of

the other diagonal .

Commented by Tinkutara last updated on 20/Jun/17

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 24/Jun/17

B o \times o M . s i n (B o M) = D o \times o M . s i n (180 - B o M) \Rightarrow

\Rightarrow B o = D o \Rightarrow A C p a s s e s f r o m m i d p o i n t

o f B D .

Commented by Tinkutara last updated on 27/Jun/17

This is incorrect solution . Should I

post the correct answer ?

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/Jun/17

[B C M] = [C M D] \Rightarrow B M = M D

[B M A] = [D M A] \Rightarrow ∠ B M A = ∠ D M A

\Rightarrow {\begin{cases} B M = M D \\ ∠ B M o = ∠ D M o \end{cases} \overset{o M = o M}{\Rightarrow} Δ B M o = Δ D M o

\Rightarrow B o = D o \Rightarrow A C p a s s e s t r o u g h m i d p o i n t

o f B D .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 27/Jun/17

y e s o f c o r s e . p l e a s e p o s t i t .

Commented by Tinkutara last updated on 27/Jun/17

Commented by Tinkutara last updated on 27/Jun/17

This is my {book}^{'} s solution . Please

explain the last line .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 27/Jun/17

w e p r o v e t h a t : D o = B o, a n d i t i s

p r o v e s t h a t A C p a s s e s t r o u g h m i d p o i n t

o f B D .