SLC S23 Week3 || Geometry with GeoGebra: Circles and its elements

_{Image taken from Pixabay}

Well, look at that—you just rolled right into my post! Today, I’m presenting the homework from @sergeyk's geometry class, so if you're feeling well-rounded you can find it here:
SLC S23Week3 || Ge◯metry with Ge◯Gebra: Circle, Its Elements, and ◯thers!

Construct a circle and show its main elements.

The main elements of the circle are:

• The Center which represents a fixed point located inside the circle from which all the other points on the circle are equidistant. Noted as O

• Radius is a segment connecting the center of the circle to any other point located on it. Noted as OA

• The Diameter is defined as a segment that connects 2 points located on the circle through the center of the circle. It is also twice the length of the radius. Noted as BC

• A Chord is a simple segment that connects any 2 points on the circle. The diameter is the longest chord in a circle. Noted as DE

• The Arc can be considered the portion on the circle between 2 points. Is also a portion of the circumference. There are 2 types: Minor Arc that is less than half of the circle and the Major Arc that is more than half of the circle. Between the points F and G

• The Circumference is the total length of the circle given by the formula: C = 2πr

• The Tangent is a straight line that has only one point of contact with the circle and is perpendicular to the radius at that point. Point H

• The Sector is the region between 2 radii and the arc between them. Noted as IOJ

• The Segment represents a section contained by a chord and the arc it created

• The Secant is the line that passes the circle in 2 points

An overview with all the elements presented above:

Construct circle O and chord AB on it. On this chord, construct the central angle. Place point D on the circle, and construct the inscribed angle ∠ADB on the same chord AB. Show the degree measures of these two angles (∠AOB and ∠ACB). Move point D. What conclusion can be drawn?

Let's first start with the circle and the chord highlighted in purple. I also added the a point D in orange on the circle.

Next, I constructed the 2 angles: ∠AOB in green and ∠ADB in orange:

Now, before we even move point D on the circle, we see that ∠AOB angle is twice the ∠ADB angle. When moving the said point, the value of the angles remain the same but when moving A or B the angle value changes. But ∠AOB is always twice ∠ADB:

If I remember correctly from my high school years, the ∠ADB angle can be called an inscribed angle as its formed by 2 chords: AD and DB.

Construct segment CD. On this segment CD, as the diameter, construct a semicircle. Take point F on the semicircle and form the angle ∠CFD. Move point F. What conclusion can be drawn?

I started with constructing the segment and the semicircle and added the point F on it. After adding the angle, I did a bit of styling just so it will stand out more.

By moving the point F, we can see that the measure of ∠CFD angle will always be 90^o, therefore its a right angle.

Construct triangle ABC. Construct two circles – the inscribed and circumscribed circles. (Do not use the "circle through three points" tool for the circumscribed circle.)

• Inscribed Circle represents the circle inside a polygon, in our case the triangle ABC, that touches each side in exactly one point

To do this, I started with constructing the triangle using the polygon tool, added 2 of the angle bisectors, one for ∠ABC and one for ∠ACB and marked their intersection point with E.

_{Starting point}

The next step is to draw a perpendicular line on BC segment that passes through the intersection point of the 2 bisectors and marked that intersection with J. I also added the angle just to be sure.

At this point in the task, I added a segment from point E to J which represents the radius of the inscribed circle we need to draw.

And lastly, its as simple as just selecting the Circle with Center tool and select E as the center:

_{Live construction of the Inscribed Circle}

With all the auxiliary lines hidden, we get a clearer view of the circle:

• Circumscribed Circle is the circle that passes to all the vertices of the polygon.

Yet again we start with constructing a ABC triangle but this time construct 2 perpendicular bisectors of any 2 sides and mark their intersection point with O:

I also added the segment OA, which will represent the radius of the circumscribed circle just like OB or OC, and marked it with red.

Just like for the inscribed circle, we can use the Circle with Center tool, select O as the center and A and the 2nd point:

_{Circumscribed Circle construction}

Use your imagination – construct something similar to this. Do you know what it is called?
As an alternative, it is possible to depict the arrangement of two (three) circles. They intersect/do not intersect or touch. Show the touch as internal and external.

The 2 constructions presented are variations of the Steiner Chain. The Steiner Chain represents a group of n circles where each one of them is tangent to 2 specific circles that do not intersect. The second one looks to be a Steiner Chain with a Triangle.

Unfortunately, neither my imagination or my GeoGebra skills can come up with something similar to these 2 constructs, so I will have to choose the alternative.

So, let's see some 3 circle arrangements:

1st Case

• we have an intersection between Circle 1 and Circle 2
• Circle 1 has internal touch with Circle 3
• Circle 2 has internal touch with Circle 3

2nd Case

• Circle 1 external touch with Circle 3
• Circle 2 internal touch with Circle 3
• No intersections present

3rd Case

• Circle 1 external touch with Circle 3
• Circle 2 external touch with Circle 3
• No intersections present

4th Case

• Intersection between Circle 2 and Circle 3
• Intersection between Circle 3 and Circle 1
• Intersection between Circle 2 and Circle 1
• All 3 circles share a common point, M

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This week's homework was a bit easier than the last one but enjoyable nonetheless. I would like to invite @khursheedanwar since we held a class on Algebra, maybe he will like this.

See you next time!