In the mini question, why can't it be AE x AF= BC squared?

[The Blade Dancer]

Module 2 Week 2 Day 8 Your Turn Part 2

In the mini question, why can't it be AE x AF= BC squared?

[debbie]

@The-Rogue-Blade This is a good question! If we take the Power of a Point Theorem literally, then the segments that we are multiplying together should live up to the name "Power of a Point"; they should all emanate from the same point.

In the above illustration, the correct answer is shown:

$$ CF \times CE = (BC)^2 $$

Now consider \(\overline{AE}\) and \(\overline{AF};\) these segments emanate from point \(A,\) but the segment \(\overline{BC}\) emanates from a \(\textcolor{red}{\text{different}}\) point, \(C.\) We have not one point, but \(2\) points.

Unfortunately, it's not called "Power of Two Points," so

$$ AE \times AF \neq (BC)^2 $$