Talk:Intersecting chords theorem

Problem with unclear terminology

The article's last sentence is the following:

"Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chords theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem."

But following that link to the article "Power of a point" finds many theorems mentioned there, but none with the name "power of point theorem" or even any similar name.

So it is not clear what this sentence is trying to say. — Preceding unsigned comment added by 2601:200:c082:2ea0:707c:50e9:cb80:af56 (talk) 16:14, 20 December 2023 (UTC)

These three theorems listed (the "tangent–secant", "intersecting secants", and "intersecting chords" theorems) are all simple consequences of the "power of a point", as described in some detail at power of a point. For a given point and circle, given any arbitrary oriented line through the point intersecting the circle, the product of the signed distances from the given point to two intersections of the line and the circle is an invariant, the "power" of the point with respect to the circle.

Alternately, you can think of the three listed theorems as expressions of the same idea as the "power of a point" idea, but using the tools and methods of ancient Greek geometry. –jacobolus (t) 07:08, 16 January 2024 (UTC)

Chord to chord theorem

Chord to chord theorem 120.28.164.226 (talk) 22:46, 15 January 2024 (UTC)

You're going to have to be clearer. I don't understand what you are getting at. –jacobolus (t) 07:04, 16 January 2024 (UTC)

wobbly reference to the power-of-a-point theorem

In the passage "Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chords theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - the power of point theorem." the 'a' is missing: "... - the power of a point theorem", but the deeper problem is that in the Wikipedia article linked to, the theorem is not cleanly stated: only the concept of the power of a point is discussed.

An example where the theorem is cleanly stated is given in the article 'Power of a Point Theorem' in the AoPS website: https://artofproblemsolving.com/wiki/index.php/Power_of_a_Point_Theorem Kontribuanto (talk) 14:56, 24 May 2024 (UTC)

Pythagoras’s Theorem

It’s easy enough to prove ab=cd as if we express the radius in two different ways using Pythagoras’s theorem and equate the expressions we end up with this result. If we can find a reliable source using this proof we should add it. It’s more obvious when the chords are intersecting and perpendicular in which case we can prove the perpendicular chord theorem (a^2+b^2+c^2+d^2=4r^2) and the intersecting chords theorem (ab=cd) using the same expression for r^2 (namely r^2=((b-a)/2)^2+((d+c)/2^2)=((b+a)/2)^2 +((d-c)/2)^2.Overlordnat1 (talk) 13:27, 27 January 2026 (UTC)