MHB Prove: (AC)² = (AD)² + (AB)·(CD)

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In trapezoid ABCD, where AB is parallel to CD and AD equals BC, the relationship AC² = AD² + (AB)(CD) needs to be proven. The proof involves applying the properties of trapezoids and the Pythagorean theorem. By constructing perpendiculars from points A and B to line CD, the segments can be analyzed to demonstrate the equality. The geometric relationships and congruencies established in the trapezoid facilitate the proof. This confirms the stated equation holds true under the given conditions.
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A trapezoid ABCD ,AB//CD ,AD=BC

Prove :

$AC^2=AD^2+(AB).(CD)$
 
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Re: Prove :AC^2=AD^2+(AB).(CD)

Albert said:
A trapezoid ABCD ,AB//CD ,AD=BC

Prove :

$AC^2=AD^2+(AB).(CD)$

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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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