Basic Geometry Inequalities!

Geometry Level 4

Let A B C ABC be a triangle such that A B = 7 , AB=7, and let the angle bisector of ∠ B A C \angle BAC intersect line B C BC at D . D. If there exist points E E and F F on sides A C AC and B C BC respectively, such that the lines A D AD and E F EF are parallel and divide △ A B C \triangle ABC into three parts of equal area, determine the number of possible integer values for B C . BC.


The answer is 13.

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Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Shaun Leong
Nov 26, 2016

By angle bisector theorem B D : D C = 1 : 2 ⇒ A B : A C = 1 : 2 BD:DC=1:2 \Rightarrow AB:AC=1:2 ⇒ A C = 14 \Rightarrow AC=14

By the triangle inequality 14 − 7 < B C < 14 + 7 14-7 < BC < 14+7 8 ≤ B C ≤ 20 8 \leq BC \leq 20 ⇒ 13 values \Rightarrow \boxed{13} \mbox{values}

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