Classroom action: Student declares Angle Bisectors to be Perpendicular Bisectors

Some students seem to be especially good at following directions–and as Lockhart asserts, math is about finding new directions not following directions–but today, one student (A) whom I’d assumed was that way proved me wrong.
The lesson was about how to construct an angle bisector, and I’d left it up to the kids to muck about for a while to figure it out for themselves. So then student A’s partners figure out how to construct it and explain it to the class.
Student A asserts that an angle bisector is a perpendicular bisector.
Nah, I say, they do 2 different purposes. A PB bisects a segment, and an AB bisects an angle.
No, look, student A patiently tells me. If you connect the 2 points on the angle, then the Angle Bisector is a Perpendicular Bisector.
And I look, and it is.  There’s a chord on the arc, and its perpendicular bisector is the same as the angle bisector.

I don’t say it today, but when it comes time to find the center of a circle, I know student A is going to have an easy time of it.

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