Proving AD Bisects Angle CAE in Triangle ABC

  • Thread starter Michael_Light
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    Geometry
In summary, the conversation discusses how to prove that AD bisects angle CAE in a triangle ABC with BC=2AB. The suggested approach is to use the cosine rule and compare the cosines of the angles in triangles AED and ADC, as well as in triangles ABE, ABD, and ABC. This can also be done using dot products or components of vectors and trigonometry.
  • #1
Michael_Light
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Homework Statement



Given a triangle ABC with BC = 2AB. D and E are the midpoints of BC and BD respectively. Show that AD bisects angle CAE.

Homework Equations





The Attempt at a Solution



Let AB= x, so DC= AB= x and ED = x/2. If AD bisects angle CAE,
then AC/AE = DC/DE
AC/AE = x/(x/2)
AC: AE = 2:1

That's how far i can do... I tried using sine rule to prove that AC:AE = 2:1 but it did not help. Can anyone guide me?
 
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  • #2
Can anyone help me? =(
 
  • #3
Hi Michael_Light! :smile:

You might try the cosine rule on triangles AED and ADC with respect to the angles you're interested in.
Followed by the cosine rule on triangles ABE, ABD, and ABC with respect to the angle at B.

Find the cosines of the 2 angles you're interested in and you should find they are the same...
 
  • #4
If you know about dot products, this problem becomes easy. If not, you could do it with components of vectors and trigonometry, but not without some difficulty. But to do it like that, place A at (0,0), D at (4,0), and go from there.
 

Related to Proving AD Bisects Angle CAE in Triangle ABC

What is the goal of proving AD bisects angle CAE in Triangle ABC?

The goal of proving AD bisects angle CAE in Triangle ABC is to show that the line segment AD divides the angle CAE into two equal angles. This is an important property of triangles and can be used to solve various mathematical problems.

What are the steps involved in proving that AD bisects angle CAE?

The steps involved in proving that AD bisects angle CAE are:

  1. Draw Triangle ABC and extend the line segment AD to meet the opposite side at point F.
  2. Assume that AD bisects angle CAE and mark the angles formed at point A and point F as x and y respectively.
  3. Use the Angle Sum Property of Triangles to show that x+y equals angle CAE.
  4. Apply the Angle Bisector Theorem to prove that x and y are equal, thus showing that AD bisects angle CAE.

What are the key theorems used to prove that AD bisects angle CAE?

The key theorems used to prove that AD bisects angle CAE are the Angle Sum Property of Triangles and the Angle Bisector Theorem. The Angle Sum Property states that the sum of all angles in a triangle is 180 degrees. The Angle Bisector Theorem states that if a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the adjacent sides.

Why is it important to prove that AD bisects angle CAE?

Proving that AD bisects angle CAE is important because it helps us understand the properties of triangles better. It also allows us to apply the Angle Bisector Theorem in problem-solving and geometric constructions. Additionally, it is a fundamental concept in geometry and is used in more complex proofs.

What are some real-life applications of proving that AD bisects angle CAE?

Some real-life applications of proving that AD bisects angle CAE include:

  • Calculating the distance between two points using triangulation.
  • Designing and constructing structures such as bridges, buildings, and roads.
  • Creating maps and navigation systems.
  • Measuring angles and distances in surveying and land mapping.

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