finding a circle center using angle in a Semicircle (Thales' Theorem)

Hannibal lecter · 2023-07-22 02:00:12

from this : https://www.mathsisfun.com/geometry/circle-theorems.html

my problem is when I draw a circle on a paper using circle template tool and trying to do these steps on MIF :

Finding a Circle's Center
finding as circles center
We can use this idea to find a circle's center:

draw a right angle from anywhere on the circle's circumference, then draw the diameter where the two legs hit the circle
do that again but for a different diameter
Where the diameters cross is the center!

I don't know how to draw a right angel from a circle circumference
how to know this angel would be 90 degree
is there any video illustrate these steps

asadkcc · 2023-07-22 05:45:51

In the realm of geometry, the exploration of theorems and principles has led to astounding revelations about the relationships between shapes and figures. One such remarkable theorem is Thales' Theorem, an elegant concept that involves semicircles and provides a unique method for finding the center of a circle. Understanding the intricacies of Thales' Theorem not only enriches our knowledge of geometry but also finds practical applications in various fields, from engineering to architecture. In this article, we will unravel the beauty of Thales' Theorem and delve into its step-by-step method for finding the elusive circle center within a semicircle.

Thales' Theorem Explained:
To comprehend Thales' Theorem, we must delve into the geometric insight of the ancient mathematician Thales of Miletus. The theorem states that if a triangle is inscribed in a semicircle, and one of its vertices lies on the semicircle's diameter, then the angle opposite the diameter is a right angle. This fundamental property of a semicircle forms the foundation of Thales' Theorem, revealing the inherent connections between angles and circles.

Proof of Thales' Theorem:
Thales' Theorem can be demonstrated through various geometric proofs, each showcasing the elegance of this principle. One method involves utilizing triangle similarity to establish the right angle. Another approach employs the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half the measure of the intercepted arc. Combining these insights reveals the equality of angles, ultimately proving Thales' Theorem.

Step-by-Step Method to Find the Circle Center:
Applying Thales' Theorem to find the circle center within a semicircle involves a systematic approach. Begin by identifying the semicircle in question and selecting three points on its circumference. Then, construct triangles with these points and the center of the semicircle. By carefully examining the angles formed, you can deduce the coordinates of the circle center.

Applying Trigonometry Techniques:
Incorporating trigonometric functions can provide an alternative means to find the circle center. By employing trigonometric ratios and working with right-angled triangles, you can establish relationships between the circle center and the points on the semicircle. This approach offers a powerful tool for solving intricate problems involving semicircles and angles.

Practical Examples and Applications:
The versatility of Thales' Theorem extends beyond theoretical proofs. It finds practical applications in real-world scenarios, ranging from architectural designs to engineering blueprints. Understanding Thales' Theorem empowers problem-solving in geometry, enabling individuals to tackle complex configurations with confidence.

Advanced Concepts and Variations:
Building on the foundation of Thales' Theorem, mathematicians have explored advanced concepts and variations. These extensions include applying the theorem to inscribed circles, exploring circles within circles, and even venturing into higher dimensions, revealing the depth of its implications.

Thales' Theorem and History:
The origins of Thales' Theorem date back to ancient Greece, attributed to the mathematician Thales of Miletus. Its impact on the development of mathematics and geometry has been profound, shaping the way we understand spatial relationships. Today, Thales' Theorem continues to be an integral part of geometry education.

Challenges and Limitations:
While Thales' Theorem is a powerful tool, it also comes with certain limitations. Euclidean geometry, on which the theorem is based, has its constraints, especially when dealing with complex semicircle configurations. Computational techniques and precision play a significant role in addressing these challenges.

Conclusion:
In the realm of geometry, Thales' Theorem stands as a testament to the elegance and insight of ancient mathematicians. Its application in finding the circle center within a semicircle opens up a world of geometric exploration. From practical applications in architecture and engineering to its influence on mathematical history, Thales' Theorem continues to captivate minds and inspire mathematical discovery. Embrace the beauty of Thales' Theorem and unlock the hidden secrets of semicircles, witnessing the harmony between angles and circles in a timeless dance of geometry.

Bob · 2023-07-25 00:43:34

Here's a way to find the centre by construction. You'll need a sharp pencil, a ruler and a compass (for drawing circles).

In this diagram the starting point is the large green circle.

Draw any line that cuts the circle in two places, A and B. Extend the line at the B end.

With B as centre draw a smaller circle that cuts AB at C and D.

Set the compass to a larger radius; then with C as centre draw two arcs judging by eye where to make them**. Repeat from D so that these second arcs cut the first. Where these arcs cross gives the points E and F. ** So that the second arcs cross at E and F.

Join E to F and extend this line to cross the original circle at G.

Line EFG is perpendicular to AB. (You can easily prove this by considering the quadrilateral CFDE and showing it is a rhombus.)

So ABG is 90 and so AG is a diameter.

Repeat this process with a new line A'B' and construct a new diameter A'G'.

Where the two diameters intersect is the centre.

Bob

Hannibal lecter · 2023-07-30 01:47:48

Bob wrote:

Here's a way to find the centre by construction. You'll need a sharp pencil, a ruler and a compass (for drawing circles).
https://i.imgur.com/ve024Du.gif
In this diagram the starting point is the large green circle.
Draw any line that cuts the circle in two places, A and B. Extend the line at the B end.
With B as centre draw a smaller circle that cuts AB at C and D.
Set the compass to a larger radius; then with C as centre draw two arcs judging by eye where to make them**. Repeat from D so that these second arcs cut the first. Where these arcs cross gives the points E and F. ** So that the second arcs cross at E and F.
Join E to F and extend this line to cross the original circle at G.
Line EFG is perpendicular to AB. (You can easily prove this by considering the quadrilateral CFDE and showing it is a rhombus.)
So ABG is 90 and so AG is a diameter.
Repeat this process with a new line A'B' and construct a new diameter A'G'.
Where the two diameters intersect is the centre.
Bob

Mr Bob I want to find the center according to MIF using thales theorm only

Bob · 2023-07-30 02:36:48

Wiki wrote:

Thales's theorem
Wikipedia
https://en.wikipedia.org › wiki › Thales's_theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle.

So that is what I used.

Bob

Math Is Fun Forum

#1 2023-07-22 02:00:12

finding a circle center using angle in a Semicircle (Thales' Theorem)

#2 2023-07-22 05:45:51

Re: finding a circle center using angle in a Semicircle (Thales' Theorem)

#3 2023-07-25 00:43:34

Re: finding a circle center using angle in a Semicircle (Thales' Theorem)

#4 2023-07-30 01:47:48

Re: finding a circle center using angle in a Semicircle (Thales' Theorem)

#5 2023-07-30 02:36:48

Re: finding a circle center using angle in a Semicircle (Thales' Theorem)

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