What Are the Proofs for Triangle Properties in Euclidean Geometry?

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In summary: What is the relationship between the length of BC and the length of DC?In summary, the first question is asking to prove that the external bisector of an angle in a non-isosceles triangle divides the opposite side into two segments proportional to the adjacent sides. To understand this better, a diagram can be drawn with the triangle and its external bisector. 2. Prove that two vertices of a triangle and the feet of the altitudes to the sides adjacent to the third vertex can be inscribed in a circle. (The feet are the points of intersection of the altitude with the opposite sides of the triangle).To prove this, draw a triangle ABC and construct the altitudes from each vertex to the opposite side
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k3N70n
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Euclidean Geometry

Homework Statement




I'm not looking for any answer, I'm just having a hard time understanding what these two questions are asking for:

1. Prove that the external bisector of an angle of a triangle (not isosceles) divides the opposite side (externally) into two segments proportional to the sides of the triangle adjacent to the angle.


2. Prove that two vertices of a triangle and the feet of the altitudes to the sides adjacent to the third vertex can be inscribed in a circle. (The feet are the points of intersection of the altitude with the opposite sides of the triangle).

If someone would help me with a picture or something that would be ideal. Thank you kindly,
Kenton.
 
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1. Prove that the external bisector of an angle of a triangle (not isosceles) divides the opposite side (externally) into two segments proportional to the sides of the triangle adjacent to the angle.


Draw a triangle ABC.

Extend the line segment AB through A. Let D be a point on that extension.

CAD is an external angle at A, is it not?

Draw the angle bisector of CAD.

Extend side BC.

The line BC intersects the angle bisector of CAD, does it not?
 

1. What is Euclidian Geometry?

Euclidian Geometry is a branch of mathematics that studies the properties and relationships of points, lines, angles, and shapes in a flat, two-dimensional space. It is named after the ancient Greek mathematician Euclid, who wrote the influential book "Elements" that presented the fundamental principles and theorems of this geometry.

2. What are the basic principles of Euclidian Geometry?

The basic principles of Euclidian Geometry include the concepts of points, lines, and planes, as well as the axioms and postulates that govern the relationships between these elements. The most fundamental postulate is the Parallel Postulate, which states that given a line and a point not on that line, there is only one line that can be drawn through the point parallel to the given line.

3. What are the main differences between Euclidian and Non-Euclidian Geometry?

The main differences between Euclidian and Non-Euclidian Geometry lie in the treatment of the Parallel Postulate. While Euclidian Geometry assumes that the postulate is true, Non-Euclidian Geometry explores the possibility of different geometries where the postulate does not hold. In Non-Euclidian Geometry, the sum of the angles in a triangle may not add up to 180 degrees, and parallel lines may intersect.

4. How is Euclidian Geometry used in real life?

Euclidian Geometry has numerous real-life applications in fields such as architecture, engineering, and physics. It is used to design buildings, bridges, and other structures with precise measurements and angles. It is also used in navigation, as well as in the study of gravity and motion.

5. What are some famous theorems in Euclidian Geometry?

Some of the most famous theorems in Euclidian Geometry include the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides; the Angle Bisector Theorem, which states that an angle bisector divides the opposite side of a triangle into segments that are proportional to the adjacent sides; and the Inscribed Angle Theorem, which states that an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc.

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