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Area of a Quadrilateral

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Question

$P r o v e t h a t e v e r y l i n e s e g m e n t h a s o n e a n d o n l y o n e m i d p o i n t .$

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Solution

$T o p r o v e : E v e r y l i n e h a s o n e a n d o n l y o n e m i d p o i n t . P r o o f : S u p p o s e C a n d C^{'} a r e t w o m i d p o i n t s o f l i n e s e g m e n t A B T h e n, A C = \frac{1}{2} A B a n d A C^{'} = \frac{1}{2} A B A C = A C^{'} (T h i n g s w h i c h a r e e q u a l t o t h e s a m e t h i n g a r e e q u a l t o o n e a n o t h e r) T h i s i s p o s s i b l e o n l y w h e n C a n d C^{'} c o i n c i d e . H e n c e e v e r y l i n e s e g m e n t h a s o n e a n d o n l y o n e m i d p o i n t .$

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