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Questions and Answers
What does the angle bisector theorem establish?
What does the angle bisector theorem establish?
What is the ratio that is equal according to the angle bisector theorem?
What is the ratio that is equal according to the angle bisector theorem?
How can it be determined that AD is the angle bisector of angle A?
How can it be determined that AD is the angle bisector of angle A?
Which theorem outlines an alternative proof for the Angle Bisector Theorem?
Which theorem outlines an alternative proof for the Angle Bisector Theorem?
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What applications does the Angle Bisector Theorem have in geometry?
What applications does the Angle Bisector Theorem have in geometry?
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In what way does the law of sines contribute to proving the angle bisector theorem?
In what way does the law of sines contribute to proving the angle bisector theorem?
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In the Triangle Proportionality Theorem, what does the ratio of the length of side BD to CD equal?
In the Triangle Proportionality Theorem, what does the ratio of the length of side BD to CD equal?
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If the angle bisector intersects side BC at point D, according to the Triangle Proportionality Theorem, what ratio is equivalent to (BD) / (CD)?
If the angle bisector intersects side BC at point D, according to the Triangle Proportionality Theorem, what ratio is equivalent to (BD) / (CD)?
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Which side length ratio in a triangle remains equivalent to the ratio of another side length to the hypotenuse, as per the Triangle Proportionality Theorem?
Which side length ratio in a triangle remains equivalent to the ratio of another side length to the hypotenuse, as per the Triangle Proportionality Theorem?
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If an angle bisector intersects a triangle's side, which ratio is NOT used in the Triangle Proportionality Theorem?
If an angle bisector intersects a triangle's side, which ratio is NOT used in the Triangle Proportionality Theorem?
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How does the Triangle Proportionality Theorem relate the lengths of sides in a triangle when an angle bisector is involved?
How does the Triangle Proportionality Theorem relate the lengths of sides in a triangle when an angle bisector is involved?
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Which theorem specifically addresses the relationship between side lengths in a triangle connected by an angle bisector?
Which theorem specifically addresses the relationship between side lengths in a triangle connected by an angle bisector?
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What is the ratio of the length of BD to the length of CD in the Triangle Proportionality Theorem?
What is the ratio of the length of BD to the length of CD in the Triangle Proportionality Theorem?
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How is the Triangle Proportionality Theorem proved?
How is the Triangle Proportionality Theorem proved?
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What do the ratios (AB) / (BD) and (AC) / (CD) represent in the Triangle Proportionality Theorem?
What do the ratios (AB) / (BD) and (AC) / (CD) represent in the Triangle Proportionality Theorem?
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What is the relationship between angles BAD and DAC in the proof of the Triangle Proportionality Theorem?
What is the relationship between angles BAD and DAC in the proof of the Triangle Proportionality Theorem?
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What can be determined using the Triangle Proportionality Theorem in geometry?
What can be determined using the Triangle Proportionality Theorem in geometry?
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What relationship does the Triangle Proportionality Theorem establish?
What relationship does the Triangle Proportionality Theorem establish?
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Study Notes
Triangle Angle-Bisector Theorem
Introduction
The angle-bisector theorem is a fundamental concept in geometry that deals with the relationship between the angles and sides of a triangle. It establishes a proportionality between the sides of a triangle and how they are divided by the angle bisector.
Statement of the Angle Bisector Theorem
The angle bisector theorem states that if a point D lies on the side BC of a triangle ABC, the ratio of the distance from D to the endpoints of BC (i.e., BD/DC) is equal to the ratio of the lengths of the other two sides of the triangle (AC/AB). Furthermore, if a point D divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle A.
Proof of the Angle Bisector Theorem
There are multiple ways to prove the angle bisector theorem. One proof can be found in the text of "Alternative Proofs for the Length of Angle Bisectors Theorem on Triangle". Another proof using the law of sines is outlined in the search result, where it is shown that the ratio of the distances from D to the endpoints of BC is equal to the ratio of AC and AB.
Applications of the Angle Bisector Theorem
The angle bisector theorem has various applications in geometry, such as proving the coordinates of the incenter of a triangle and helping to understand circles of Apollonius. It is also used in other theorems and results, such as proving the coordinates of the incenter of a triangle.
Alternative Proofs
The search results mention several alternative proofs for the standard lengths of angle bisectors and the angle bisector theorem in any triangle, which are elaborated in the cited papers. These alternatives offer different ways to understand and apply the theorem in various contexts.
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Description
Learn about the angle-bisector theorem, a fundamental concept in geometry, which establishes a proportionality between the sides of a triangle and how they are divided by the angle bisector. Explore the statement, proofs, applications, and alternative proofs of this theorem.
