Geometry Fireballs!

Geometry Level 2

ABCD is a convex quadrilateral such that A B 2 + C D 2 = A D 2 + B C 2 AB^2+CD^2=AD^2+BC^2 what can we say about AC and BD?

AC+3AB=2BD AC=BD AC⊥BD AC+AB=BD

Mohammed Imran by Mohammed Imran , 13, India

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1 solution

Let the position vectors of A , B , C A, B, C and D D relative some reference system be a → , b → , c → \overrightarrow a, \overrightarrow b, \overrightarrow c and d → \overrightarrow d respectively. Then the given equation yields ( b → − a → ) 2 + ( d → − c → ) 2 = ( d → − a → ) 2 + ( c → − b → ) 2 (\overrightarrow b-\overrightarrow a) ^2+(\overrightarrow d-\overrightarrow c) ^2=(\overrightarrow d-\overrightarrow a) ^2+(\overrightarrow c-\overrightarrow b) ^2 . Simplifying we get ( c → − a → ) ⋅ ( d → − b → ) = 0 (\overrightarrow c-\overrightarrow a) \cdot (\overrightarrow d-\overrightarrow b) =0 or A C → ⋅ B D → = 0 \overrightarrow {\rm AC}\cdot \overrightarrow {\rm BD}=0 . Hence A C → ⊥ B D → \boxed {\overrightarrow {\rm AC}\perp \overrightarrow {\rm BD}}

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Good solution

Mohammed Imran - 1 year, 3 months ago

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