Power of a point and some fun equalities

Way back in 2011 when I first started thinking about doing math movies I filmed a set of practice lectures based on the first couple of sections of Geometry Revisited. The main point was to evaluate myself talking about math – at that point it had been more than 10 years since I’d been in a classroom. Thefirst practice lecture where I try to shake of some of the old rust was about the extended law of sines.

What I learned in my trig class back in high school was that for a triangle: a / Sin(A) = b / Sin(B) = c / Sin(c). That’s an incredible identity, but there’s a pretty natural question that I never thought to ask – if these expressions all have the same value, is there something special about that value? That’s the question that this lecture tries to answer:

Today I went through a similar exercise with my son as we worked through the review section in the Power of Point chapter in our geometry book. Studying the power of a point identities you learn that lots of products involving lines and circles are equal to each other – but does the value of the product have any special significance? Let see . . . .

So, for points outside for the circle we just found the value of the products that come up in the power of a point formulas. We saw, though, that when the point is inside of the circle we’ll need to have a different formula. We derive that companion formula here:

So, a fun trip down memory lane for me this morning while reviewing some power of a point ideas. Hopefully work like this plants a little mathematical idea in my son’s mind – it two (or more) expressions are equal to each other, maybe there’s something special about the value of those expressions.