Finding the length of a triangle given one side and the ratio the median and angle bisector cut the altitude.

Question

A median BK, an angular bisector BE and an altitude AD are drawn in a triangle ABC. Find the side AC if it is known that the line BK and BE divide the line segment AD into 3 equal parts and if AB = 4cm. Find the length of AC.

Currently, I am able to work out that BD=2 by angle bisector theorem and that,since,sin BAD= 1/2; BAD=30 and angle ABC=60 but after that I'm unable to work it out further.

Answer 1

I can help you with the construction, and you should be able to do the rest. Forget about the length of $AB$ for now.

1. Draw altitude $AH$ , draw the baseline (where points $B$ and $C$ should lie), and draw the midline (where point $K$ should lie). Mark points $X$ and $Y$, which divide $AH$ into 3 equal segments.

2. Draw circle $c$ with center $X$ and radius $XH$.

3. $X$ is on the bisector of $\widehat{ABC}$, so $AB$ should be tangent to circle $c$. Draw $AB$.

4. Draw $BY$. It meets the midline at $K$, the midpoint of $AC$.

5. Draw $AK$ and extend it to meet the baseline at $C$.

Now, can you do the rest?