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Inscribe: To draw one figure within another figure so that every vertex of the enclosed figure touches the outer figure (Free Dictionary). 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).

From the definition of “inscribed”, it follows that the two endpoints of the diameter of the semi-circle must lie on the edges of the square and that we are looking for the largest such semi-circle that lies within the square. We start by exploring whether these two points lie on the same, on opposite or on adjacent sides of the square.

TEST 1. Create two dynamic points E and F on the base of the square and a circle with diameter d=|EF|.

Click once and then again on the base of the square.

Point E

Point F

j=Segment[E,F]

Click once on segment j to get midpoint

Point G=Midpoint[j]

Click once on G and then on E and then on G again

k=circle with center G and radius {G,E]

Exploring: Move E and F along base. Is the semi-circle always inscribed? What is the maximum diameter possible for EF?

TEST 2. Create two dynamic points E on the base and F on the top of the square and a circle with diameter d=|EF|.

Reset your GeoGebra drawing to just the square.

(or use undo )

Click once on the base and then again on the top of the square.

Point E

Point F

j=Segment[E,F]

Click once on segment j to get midpoint

Point G=Midpoint[j]

Click once on G and then on E and then on G again

k=circle with center G and radius {G,E]

Exploring: Move E and F along base. Is the semi-circle always inscribed? When is it inscribed? What is the maximum diameter possible for EF when the semi-circle is inscribed?

TEST 3. Create two dynamic points E on the base and and F on the (left) side of the square and a circle with diameter d=|EF|.

Reset your GeoGebra drawing to just the square.

(or use undo )

Click once and then again on the base of the square.

Point E

Point F

j=Segment[E,F]

Click once on segment j to get midpoint

Point G=Midpoint[j]

Click once on G and then on E and then on G again

k=circle with center G and radius {G,E]

Exploring: Move E and F along base. (1) What do you notice about the circle and the vertex A of the square? Change the square (move A or B). Does this property still hold?

(2) Continue to move E and F until you get the largest inscribed semi-circle. Can you determine the maximum diameter possible for EF? (The answer to this last question is probably no - at least not without some more exploring and thinking.) But can you say anything about the points and the line segments.

Some about what to look for and think about.

• Can you say anything about |EA| and |FA|? • Can you say anything about the point G and the diagonal AC? • Can you say anything about the diameter EF and the diagonal AC?

• What line segments are radii of the semi-circle?

3) Step 3: Finding the properties of the inscribed semi-circle

Reset your GeoGebra drawing to just the square.

(or use undo )

We want to construct a semi-circle with the properties we found in step 2. We need the diagonal AC, a point G on the diagonal and the intersection points E and F of the line through G that is normal (perpendicular) to AC. Then we will click and drag G until we get an inscribed semi-circle and see what else we can see.

Click once on vertex A and then again on vertex C to get the diagonal of the square.

Segment: j=Segment[A,C]

Click once on the diagonal j to get a movable point. Rename it G.

Point: G=Point[j]

Click once on G and then on diagonal j to get normal (perpendicular).

Line: k=Perpendicular[G,j]

Click on k and then on base of square f to get point E.

Repeat with k and left side of square i to get point F.

E=Intersection[k,f]

F=Intersection[k,i]

Click once on G and then on E and then on G again.

p=circle with center G and radius {G,E]

Exploring: Move G along the diagonal until semi-circle is inscribed. Look at the points where the semi-circle touches the square. Do they have any connection to G?

Draw the lines through G perpendicular (normal) to the sides of the square. Click on G and then on the base f of the square. Click on G and then on the side i of the square.

Line: l=Perpendicular[G,f]

Line: m=Perpendicular[G,i]

Click on the intersection points of the normals and the sides of the squares (starting on the base and going anti-clockwise).

H=Intersection[l,f]

I=Intersection[m,g]

J=Intersection[l,h]

K=Intersection[m,i]

Exploring: Look for radii of the circle. Look for congruent triangles.

4) Step 4: Finding the construction key

With our drawing we can see that:

To do the construction, we need a length for AH or AG or AE. (Any one of these determines the semi-circle.)

The important thing to remember is that we can only measure (using our compass) the side of the square a and the diagonal AC.

We can see that triangle AGE is congruent to triangle GJC

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