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.


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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
$[BCM]=[CMD]⇒BM=MD [BMA]=[DMA]⇒∠BMA=∠DMA ⇒ { ((BM=MD)),((∠BMo=∠DMo)) :}⇒^(oM=oM) ΔBMo=ΔDMo ⇒Bo=Do⇒AC passes trough midpoint of BD.$
$[B C M] = [C M D] \Rightarrow B M = M D$ $[B M A] = [D M A] \Rightarrow ∠ B M A = ∠ D M A$ $\Rightarrow {\begin{cases} B M = M D \\ ∠ B M o = ∠ D M o \end{cases} \overset{o M = o M}{\Rightarrow} Δ B M o = Δ D M o$ $\Rightarrow B o = D o \Rightarrow A C p a s s e s t r o u g h m i d p o i n t$ $o f B D .$
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

$w e p r o v e t h a t : D o = B o, a n d i t i s$ $p r o v e s t h a t A C p a s s e s t r o u g h m i d p o i n t$ $o f B D .$
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