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Simple Approach to this Problem Please!

Hello Brilliant !
i have been solving this geometry problem before but i can't find a better and simpler approach. I tried using some analytic geometry tools but i find it complicated and "very long". Guys, can you please give me a simple and better solution to this problem?...Thanks!
Here it is: In triangle ABC, AB=10, BC=18 and AC=12. Let E be a point on BC such that AE is the angle bisector of angle BAC. Find EC.

Note by Jayali Phi
3 years, 9 months ago

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Hi, I suggest using the angle bisector theorem, which states that in this triangle the ratio of BE to EC is equal to the ratio of AB to AC. If you don't want to assume this theorem then you can obtain the same result by using the sine rule on triangle ABC to obtain the ratio SinB to SinC and equate it to the same ratio obtained by applying the sine rule separately to triangles ABE and AEC and then equating using the sine of the common angle (half of angle BAC). Lucy Kyte · 3 years, 9 months ago

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Angle bisector theorem would do the job. Sabab Ahad · 3 years, 9 months ago

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