Intersecting Secants Theorem

This is the idea (a,b,c and d are lengths):

Intersecting Secant Lines

And here it is with some actual values (measured only to whole numbers):

Intersecting Secants Lines example

And we get

Very close! If we measured perfectly the results would be equal.


Why not try drawing one yourself, measure the lengths and see what you get?


The lines are called secants (a line that cuts a circle at two points).

This also works if one or both are tangents (a line that just touches a circle at one point), but since two lengths are identical we don't write c×d or c×c we just write c2:

Intersecting Secant and Tangent Lines example

(Question: What happens when both are tangents?)


Why is this true?

Because there are similar triangles! Looking below:

  • They both share the angle θ
  • They both have the same angle φ (see inscribed angles)

Intersecting Secant and Tangent Lines reason

The triangles may not be the same size, but they have the same angles ... so all lengths will be in proportion!

Looking at the lengths coming from point "P", one triangle has the ratio a/d, and the other has the matching ratio c/b:

a/d = c/b

a × b = c × d


15601, 15602, 15603, 15604, 15605, 15606, 15607, 15608