Inequality with Circle and Triangle in Euclidean Geometry

In summary, the question involves geometry with circles and triangles. The individual asking for help is stuck on two parts of the solution and is looking for assistance. A picture is provided, along with a question about how a certain implication is true. The response explains that the implication is true by observing that certain angles are equal, and that the statement itself is not necessary to solve the problem.
  • #1
seniorhs9
22
0

Homework Statement



Please see below...

Homework Equations



Please see below...

The Attempt at a Solution



Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution...

Thank you...

http://img256.imageshack.us/img256/9475/gtewp249no24.jpg

[ How did they get this? They never explained or proved this and it is NOT obvious from the picture.

Blue: How is that implication true? By what theorem or reasoning?
 
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  • #2
To see that the red statement is true, observe that [itex]\angle[/itex]ADO = [itex]\angle[/itex]CDB > [itex]\angle[/itex]BOD by Euclid I.32 (book I proposition 32).

In fact, that statement is superfluous. Once you know that [itex]\angle[/itex]ADO > [itex]\angle[/itex]BDO = [itex]\angle[/itex]ADC = [itex]\angle[/itex]AOD + [itex]\angle[/itex]OAD, you know that [itex]\angle[/itex]ADO > [itex]\angle[/itex]OAD, and by Euclid I.19, OA > OD.
 

1. What is "Inequality with Circle and Triangle" in Euclidean Geometry?

"Inequality with Circle and Triangle" in Euclidean Geometry refers to the relationship between a circle and a triangle, where the circle is inscribed or circumscribed within the triangle. It involves comparing the lengths of the sides and angles of the triangle to the radius and diameter of the circle.

2. How is the "Inequality with Circle and Triangle" used in geometry?

The "Inequality with Circle and Triangle" is used to prove geometric theorems and solve problems involving circles and triangles. It is also used in real-world applications such as architecture, engineering, and navigation.

3. What is the difference between inscribed and circumscribed circles?

An inscribed circle is a circle that is tangent to all three sides of a triangle. A circumscribed circle is a circle that passes through all three vertices of a triangle. In other words, an inscribed circle is inside the triangle, while a circumscribed circle is outside the triangle.

4. How is the Inequality Theorem used to prove the "Inequality with Circle and Triangle"?

The Inequality Theorem states that in a triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle. This theorem is used to prove the "Inequality with Circle and Triangle" by comparing the length of the sides of the triangle to the radius and diameter of the circle.

5. What are some real-life examples of the "Inequality with Circle and Triangle" in use?

The "Inequality with Circle and Triangle" can be seen in architecture, where the relationship between a circular dome and a triangular base is used to create stable and aesthetically pleasing structures. It is also used in navigation to determine the distance between two points on a map, where the circle represents the Earth and the triangle represents the points of interest.

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