How Can We Prove Quadrilateral Similarity Through Circle Intersections?

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The discussion focuses on proving the similarity of a quadrilateral formed by the intersections of circles with diameters AB, BC, CD, and DA to the original quadrilateral ABCD. Similarity is defined as having the same angles while the side lengths differ proportionally. The participants emphasize that the lengths of the sides of the new quadrilateral scale together, maintaining the angle relationships. This geometric property underlines the concept of similarity in quadrilaterals. The proof hinges on the consistent angle measures and proportional side lengths derived from the circle intersections.
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Given a quadrilateral ABCD,
Prove that the quadrilateral formed by the intersections other than the vertices A, B, C & D
of the circles with diameter AB, BC, CD & DA
is similar to the quadrilaterar ABCD.

What is the relation of similarity?
 
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similarity means just the same up to scale hence all angles are the same but the lenghts are different.
 
Mr.Brown said:
similarity means just the same up to scale hence all angles are the same but the lenghts are different.

The lengths of the sides scale together.
 

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