Does $CD$ bisect $AE$?

Question

Construct triangle $ABC$.choose a $D$ point on the side $AB$ such that $\frac{BD}{AD}=\frac{2}{1}$.Now draw median $AE$. $AE$ intersects $CD$ at $O$ .İf area of triangle $AOD$ is 20, what is the area of triangle $ABC$? İ have tried using formula $S=\frac{1}{2}AB\cos \alpha$ for triangles $AOD$ and $AEB$ and comparing them but the side $AO$ is different than $AE$ so im so confused

Answer 1

To answer your question: Yes, $O$ is the midpoint of $AE$, and is a consequence of Menelaus' Theorem. However you do not need this to solve your question. We just need the fact that the ratio of the area of two triangles sharing the same height is equal to the ratio of the length of their bases.

Construct the segment $OB$.

Given that $AD : DB = 1:2$, the area of $\triangle BOD$ is twice that of $\triangle AOD$, i.e. its area is $40$.

Since $BE =EC$, the areas of $\triangle OEB$ and $\triangle OEC$ are equal. The areas of $\triangle AEB$ and $\triangle AEC$ are equal too. Hence the areas of $\triangle AOB$ and $\triangle AOC$ are equal by subtraction. Therefore the area of $\triangle AOC = 60$.

Now we finish off by noting that the area of $\triangle CDB$ is twice that of $\triangle CDA$.

Hence the area of $\triangle ABC = (20 + 60)\times 3 = 240$.

We can also find that the areas of the triangles $OEB$ and $OEC$ are $60$, which is equal to that of $\triangle AOC$, and gives an alternate proof that $CD$ does indeed bisect $AE$.

Answer 2

Since this question is about ratio of areas, and such ratios are preserved by linear transformations, we can choose convenient coordinates and angles for the vertices.

Choose $A=(0,0)$, $B=(1,0)$, $C=(0,1)$, $D=(\tfrac{1}{3},0)$, $E=(1,1)/2$.

Then point $O$ is the intersection of the line $y=x$ and $y=1-3x$, that is, $x=1/4=y$.

Thus the area of $AOD$ is $\tfrac{1}{2}\times\tfrac{1}{3}\times\tfrac{1}{4}=\tfrac{1}{24}$, while that of $ABC$ is $\tfrac{1}{2}$. Thus the ratio $\frac{AOD}{ABC}=\frac{1}{12}$. Since the actual area of $AOD$ is 20, that of $ABC$ is 240.