Welcome to the third week of Steemit Learning Challenge Season 23. This time, we will get to know the circle, its elements, and the positioning of the circle with some other shapes. The tasks are easier this time, but the last task is quite unusual...

The circle in geometry is no less significant than the triangle. And humanity most likely encountered the circle before the triangle – the Sun, the Moon, the trunk of a tree...

Euclid also described the circle as a set of points that are equidistant from a certain point.

This distance is called the radius of the circle, and the point is called the center of the circle. Note that the circle is a line, just a line, not the space enclosed by the line.

The first tool in GeoGebra that allows constructing a circle is Circle with Centre.

To construct it, you need to specify the center and the radius (the second point is the endpoint of the radius).

In this drawing, point A is the center of the circle (it is commonly denoted as O with a subscript if there are multiple circles), and the segment AB is the radius of the circle.

Let's place an arbitrary third point, point C, on the circle using Point on Object. The segment BC formed is called the chord. The corresponding part of the circle is called the arc BC.

The definition of a diameter can be given as follows – the diameter is the largest chord, or the diameter is the chord that passes through the center.

To construct a diameter, it is not enough to simply take two points on the circle – because this may form either a chord or a diameter. You need to take a point on the circle and the center of the circle, and then draw a line through them. This line will intersect the circle and form the other end of the diameter.

Other tools from the circle section are not as important, except for constructing a semicircle on a segment as a diameter.

Let's construct the segment AB. Then, from the 'circles' tools, we select the semicircle. First, draw from A to B, and then from B to A.

Another useful tool might be the circle construction tool with the same radius as a given circle, as shown in this image.

A straight line may not intersect the circle, may intersect the circle, or may have only one common point with the circle – it touches the circle.

In GeoGebra, there is a corresponding tool for constructing a tangent.
If you click on the circle and then on a point outside the circle, two tangent lines will be constructed.
If you click on the circle and a point on the circle, a tangent at that point will be constructed.

If a circle passes through all three vertices of triangle ABC, it is called the circumscribed circle of the triangle. In GeoGebra, there is a corresponding tool called Circle through 3 points.

However, it is necessary to know how to construct this circle without the given tool.

We construct two perpendicular bisectors of two arbitrary sides. They will intersect at point O – this point will be the center of the circumscribed circle. As we already know, the third perpendicular bisector will also pass through this point.
Next, we construct the circle with the center at point O (the intersection of the perpendicular bisectors) and passing through one of the triangle's vertices. In other words, OA, OB, and OC will be the radii of the circumscribed circle.

If the circle is tangent to all three sides of the triangle, it is called the incircle of the triangle.

Unlike the previous triangle, there is no separate tool for constructing this circle. Therefore, we construct the incircle sequentially based on the property that the center of the incircle of a triangle is the point of intersection of the angle bisectors. As you remember, you only need to construct two bisectors; the third bisector will also pass through this point.

There are also excircles – there are three of them, and they touch the sides and extensions of the other two sides. The center lies at the intersection of the angle bisector and the external angle bisector. The external angle is the one adjacent to the given angle of the triangle.

1. Construct a circle and show its main elements.

2. 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?

3. 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?

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

5. Use your imagination – construct something similar to this. Do you know what it is called?

Title of your work: SLC S23 Week3 || Geometry with GeoGebra: Circles and its elements

You can publish your work in any language, in any community, or simply in your own blog. Add the link to your work below as a comment.

To help me quickly find, review, and evaluate your work, leave the link in the comment under this text, and in your work, use the tag #gwgg-slc23w3

Each task response must include at least one image and one video (GIF) demonstrating the process of construction/solving. (if without gif, then several images explaining the process)
You can use tools like GifCam as I did.
Note: The video/GIF will have the most significant impact on the evaluation.

Plagiarism and the use of AI are prohibited.

Participants must be verified and active users of the platform.

All images used must belong to the author or be free of copyright. (Don’t forget to credit the source.)

Participants must not use any bot services for voting or engage in vote buying.

Recommend your friends to participate.

Submission Period: From Monday (March 03/2025), to Sunday (March 09/2025).

Your work will be reviewed, commented on, and evaluated by me. Four best works will be selected.