Thread: circumscribe a square about a semicircle

how do you circumscribe a square about a semicircle with a given radius?
thanks.

Originally Posted by Pabo258

how do you circumscribe a square about a semicircle with a given radius?
thanks.

I've attached a sketch.

The order of steps is:

1. black: semicircle and perpendicular bisector

2. blue: angle bisector

3. green: tangents to the semicircle

4. red: perpendicular lines through the endpoints of the diameter to the tangents of the semicircle.

Cute! I notice that actually touches the semi-circle at two points. In what sense does that "circumscribe" the semi-circle better than, say, the square that would circumscribe the entire circle, which touches the semi-circle at three points?

Originally Posted by HallsofIvy

Cute! I notice that actually touches the semi-circle at two points. In what sense does that "circumscribe" the semi-circle better than, say, the square that would circumscribe the entire circle, which touches the semi-circle at three points?

That depends on how you define "circumscribe".
As I understand it the semicircle has to determine a square. If the base line of the semicircle (the diameter of the circle) is parallel to one of the square's side then a square isn't necessary. You can take a rectangle too.

If the base line is tilted less than 45° then the semicircle defines a rectangle too, but the question asks for a square.