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Goal: a semi-circle in a square. Idea by Colin McAllister

Inscribe: To draw one figure within another figure so that every vertex of the enclosed figure touches the outer figure (Free Dictionary). In geometry, an inscribedplanarshape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. Specifically, at all points where figures meet, their edges must lie tangent. There must be no object similar to the inscribed object but larger and also enclosed by the outer figure (Wikipedia).

YouTubeMathcastorScreenCast Mathcast (if YT is blocked)

Go for it!To restart construction from dynamic square, click on the reset button at top right.

GeoGebra Construction: Watch and then explore the construction Directions for InterActivity

1. Click on Play to see the construction unfold.

2. Select the Move tool and click & drag points A and/or B. The semi-circle is always inscribed.

Proof (Colin McAllister)

An analysis of the two congruent triangles AGE and GJC is proof that we have identified the inscribed semicircle. The geometric construction can be used to calculate the radius of the semicircle inscribed in a square of side a. We have drawn a circle of radius=a centered on corner C. It intercepts the diagonal of length=a*sqrt(2) at a point Z, which is therefore a distance: a*sqrt(2)-a from corner A. By drawing a circle centered on A through Z, we identify a point H on side AB. The distance HB=AB-AH, so that HB=a-(a*sqrt(2)-a). Thus, HB=a*(2-sqrt(2)). The vertical line from H crosses the diagonal at a point G, which is the centre of the semicircle. So the radius r of the inscribed semicircle is r=a*(2-sqrt(2)). In the case of a unit square with side a=1, the radius is r=2-sqrt(2). We can also calculate the area of the inscribed semicircle, which is simply pi*r*r/2. This is the largest semicircle that fits in the square.

Note: sqrt() represents the square root function. Calculations can be represented exactly using the square root of 2, which is an irrational number, or approximately using a decimal representation of the square root of 2, to a chosen number of decimal places.

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Global

Advanced Compass and Straightedge Constructions with GeoGebra

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