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

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Hello steemians,

Here is my homework for SLC23 Week 3, with the corresponding tasks assigned by @sergeyk!

Construct a circle and show its main elements.

Arc: A portion of the circle's circumference between two points.	Center (O): The fixed middle point of the circle, equidistant from all points on the circumference.	Chord [AB]: A line segment whose endpoints lie on the circle.

Circumference: The total boundary of the circle.	Diameter [AB]: The longest chord, passing through the center; twice the radius.	Radius [OA]: A line from the center to any point on the circle; half the diameter.	Secant: A straight line that intersects the circle at two points.

Sector: A region enclosed by two radii and the corresponding arc (like a pie slice).	Segment: The region enclosed by a chord and the corresponding arc.	Tangent: A straight line that touches the circle at exactly one point.

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?

I started by drawing a circle using the "Circle with Center & Radius" tool in GeoGebra, I placed a point A as the center of the circle, although normally this point should have been noted O, then I defined an arbitrary radius to obtain a correct construction, subsequently, I traced a chord CB by selecting the "Segment" tool, placing the points C and B on the circumference of the circle, even if in the usual notation, C should have been A and B should have remained unchanged.

After that, I constructed the central angle ∠CAB by measuring the angle formed by the segments AC and AB, although this angle is normally denoted by ∠AOB, once this step was done, I added a point D on the circumference using the tool "Point", and I constructed the inscribed angle ∠DCB by measuring the angle formed by the segments DC and DB, which normally corresponds to ∠ADB if I had followed the conventional notation.

To better analyze the relationship between these angles, I displayed the measurements of the angles ∠CAB and ∠DCB using the GeoGebra options, then, I moved the point D around the circumference and I found that, regardless of its position, the inscribed angle ∠DCB remained constant and always equal to half of the angle at the center ∠CAB, thanks to this manipulation, I was able to verify experimentally the inscribed angle theorem, although I did not respect the traditional notation of points.

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 by creating a segment using the Segment tool of GeoGebra, but instead of following the requested notation and naming the points C and D, I mistakenly used the letters A and B, this involuntary modification means that I did not respect the notation expected in the statement, but this in no way affected the validity of the geometric construction carried out, subsequently, in order to precisely mark the center of this segment, I used the Middle or tool Center, which allowed me to place a point O in its center, an essential step for the construction of the semicircle that would follow.

After having defined the center of the segment, I then used the Semi-circle tool to draw a semi-circle with the diameter of the segment AB, although, in accordance with the requested notation, these points should have been called C and D, once this semi-circle was constructed, I added a point C on the curve, whereas, according to the initial instructions, this point should have been designated under the name F, after the insertion of this point on the semicircle, I traced the segments AC and BC, which made it possible to form the triangle ACB, a figure essential to the study of the angular properties of the semicircle, at this stage, I used the Angle tool in order to measure the inscribed angle ∠ACB, which is the angle formed between the segments AC and BC at the point C.

Analyzing the results provided by GeoGebra, I immediately noticed that the measurement of ∠ACB was systematically equal to 90 degrees, regardless of the position of the point C on the curve, in order to deepen this observation and verify it experimentally, I moved C (which, in the correct notation, should have been named F) along the semi-circle while monitoring the measurement of the inscribed angle, I then noticed that, whatever the position of the point on the curve, the inscribed angle remained invariably equal to 90 degrees, which concretely confirmed a fundamental property of the geometry of the circle.

Thus, this construction offered me the opportunity to highlight an immediate and unavoidable consequence of Thalès' theorem, which rigorously establishes that any angle inscribed in a semi-circle is necessarily a right angle, independently of the precise position of the point on the arc of the circle, consequently, although I did not scrupulously follow the notation required in the initial statement by assigning incorrect names to the different points of my figure, the experiment carried out nevertheless led me made it possible to confirm visually and experimentally this fundamental geometric property, thus illustrating the universality and invariance of this mathematical relationship.

Therefore, despite these unintentional modifications in the designation of the points, which in reality had no impact on the integrity of the construction, the demonstration proved to be just as relevant and valid, in fact, this experiment proved, with indisputable precision, that any angle inscribed in a semi-circle systematically measures 90 degrees, whatever the modifications made to the labeling of the points or the way in which the mobile point is moved around the circumference, this invariability highlights the robustness of geometric reasoning and the absolute nature of certain fundamental properties, which remain true regardless of formal adjustments made to the initial layout.

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

I started by creating a triangle ABC using the Polygon tool, which allowed me to define the three vertices A, B and C, once the triangle was constructed, I continued by tracing the bisectors of its three angles, which led to obtaining a single intersection point, which I named G, this point represents the center of the inscribed circle of the triangle, then, I traced a perpendicular from this point G to one of the sides of the triangle, in this case the side BC on the point D.

To construct the inscribed circle, I used the Circle with Center and Point tool, placing the center of the circle at D and taking as a second point the pointD, thus guaranteeing that the circle is tangent to the three sides of the triangle. Once this step was completed, I obtained the inscribed circle, which is the largest circle contained in the triangle and tangent to its three sides. This construction allows you to visualize the incenter, which is always located inside of the triangle and is defined as the point of intersection of the three bisectors.

After constructing the inscribed circle, I then focused on constructing the circumscribed circle, which passes through the three vertices of the triangle, unlike a direct approach with the "Circle passing through three points" tool, I followed a rigorous method by drawing the bisectors of two sides of the triangle, I selected the sides AB and BC, and I constructed their respective bisectors by drawing perpendiculars passing through their midpoints, The intersection of these two bisectors gave rise to a single point M, which is the center of the circumscribed circle.

Once the point D was found, I was able to construct the circumscribed circle using the Circle with Center and Point tool, placing the center in D and taking as a radius the distance between D and one of the vertices of the triangle (A, B or C), this allowed me to draw the circumscribed circle, which passes exactly through the three vertices of the triangle, this circle is essential in geometry since it represents one of the fundamental elements of the triangle and its center, called the circumcenter, can be located inside, outside or directly on the triangle, depending on the type of triangle studied.

By observing these dynamic constructions in GeoGebra, I was able to see that, whatever the way in which I moved the vertices of the triangle, the inscribed circle remained tangent to the three sides and the circumscribed circle always passed through the three vertices, which confirms the fundamental properties of these circles, thus, this construction made it possible to demonstrate in a visual and experimental way the importance of bisectors and bisectors in the construction of the characteristic circles of a triangle.

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.

For the first animation, the construction cannot be described as a Steiner chain, because the latter requires at least three circles inside the reference circle, here, we rather have a configuration of tangent circles, which means that each circle touches the others without respecting the precise conditions of a Steiner chain, in the second animation, we indeed find a Steiner chain inscribed in a triangle, which means that the circles are contained inside the circle circumscribed by the triangle and are tangent to one another, thus forming an elegant and symmetrical geometric structure.

My work explores both the concepts of external and internal tangency, as well as the arrangements of circles in more complex structures, this type of construction is often studied in advanced geometry and in projective geometry, notably in configurations related to Apollonius circles, Steiner chains, and tangent circles inscribed in polygonal figures.

Three separate circles with no interaction.	Two tangentially external circles, the third remains separate.	Three circles in mutual contact with marked points of tangency.

Three circles with intersections, forming a complex tangency configuration.	A circle inscribed in another and tangent to a third.	A circle is internally tangent to the other two.	Two small circles tangentially internal to a large circle.

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Thank you very much for reading, it's time to invite my friends @pelon53, @crismenia, @bossj23 to participate in this contest.

Best Regards,
@kouba01