Talk:Power of a Point Theorem

The reason the lengths are directed is so that $PA \cdot PA'$ matches the definition of Power of Point everywhere. Namely, negative inside and positive outside the circle, and zero on the circle as well. --Telliott99 (talk) 01:21, 16 November 2023 (UTC)

AFAIK, directed line segments are vectors, whose orientation and magnitude is defined with respect to two given points. In geometry, they are used mainly in the context of two parallel directed line segments. If they point in opposite directions, the magnitude is the same but signed negative. Then their product is the product of the magnitudes, signed negative. The ratio of two such segments is negative if one is signed positive and one negative. I assumed the background was on ProofWiki because I've seen them used (Menelaus's Theorem), although the definition there is not the one I'm familiar with. Since it appears the background is not on PW, I'll have to investigate the sources I have available. I suspect Mathworld may be of use as well. --Telliott99 (talk) 12:45, 16 November 2023 (UTC)

What about the other question? Is this or is this not the Power of a Point Theorem? --prime mover (talk) 12:49, 16 November 2023 (UTC)

I hesitated to make it a named theorem, since I've never seen that usage. But I'm no expert. I've seen Power of a Point with respect to a Circle as a definition due to Steiner, with these properties. The AoPS entry looks very similar but they elide the question of signed lengths. Please make it a redirect, if that's OK. It's disheartening to see all history of my work disappear with a renaming. --Telliott99 (talk) 13:05, 16 November 2023 (UTC)

Yes, it doesn't work like that. When you move a page, it moves the history as well, so that's still there for you. --prime mover (talk) 22:45, 16 November 2023 (UTC)

Never mind, I see that didn't happen here. --Telliott99 (talk) 13:07, 16 November 2023 (UTC)