Proving AD Bisects Angle CAE in Triangle ABC

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Homework Help Overview

The problem involves proving that line segment AD bisects angle CAE in triangle ABC, where BC is twice the length of AB. The midpoints D and E are defined on segments BC and BD, respectively.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to establish a ratio between segments AC and AE using the relationship between segments DC and DE, but encounters difficulties with the sine rule. Some participants suggest using the cosine rule on relevant triangles to explore the angles involved.

Discussion Status

The discussion is ongoing, with participants offering different approaches, including the cosine rule and vector components. There is no explicit consensus yet, but various methods are being explored to tackle the problem.

Contextual Notes

Participants are working under the constraints of the problem statement and are exploring different mathematical tools without a complete solution being reached.

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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Can anyone help me? =(
 
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...
 
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.
 

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