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Product of the intercepts of intersecting chords

In the circle centre O, the chords AB and CD intersect at the point E. Change the orientation and shape of the diagram by clicking and dragging any of the blue points. What do you observe?

Consider the two triangles CBE and ADE.

  1. Why is <CBE = <ADE?
  2. Why is <CEB = <AED?
  3. Using (1) and (2) we can say that the triangles CBE and ADE are similar. Why is this so?
  4. Given that the corresponding sides in similar triangles are in the same ratio, we can say that CE/AE = BE/DE. How can we use this result to prove that the product of the intersecting tangents are equal?