MHB A,B,C,D cocyclic prove AC^2=AB×AD+BC^2

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In the discussion about the cyclic quadrilateral ABCD with the condition BC = CD, the goal is to prove the equation AC² = AB × AD + BC². Participants explore geometric properties and relationships inherent in cyclic quadrilaterals, utilizing the Law of Cosines and properties of angles subtended by the same arc. The hint suggests leveraging these properties to establish the necessary relationships between the sides and diagonals. The proof ultimately relies on the equality of angles and the application of trigonometric identities. The conclusion reinforces the validity of the equation within the context of cyclic quadrilaterals.
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$ABCD$ is a Cyclic quadrilateral,given $BC=CD$

Prove:

$AC^2=AB\times AD+BC^2$
 
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Albert said:
$ABCD$ is a Cyclic quadrilateral,given $BC=CD$

Prove:

$AC^2=AB\times AD+BC^2$
hint:
using the following diagram:
View attachment 6569
 

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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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