Answer to Question #158467 in Geometry for Kelvin Sowah

Question

In a quadrilateral OABC, D is the midpoint of BC and E is the point on AD such that AE : ED = 2 : 1. Given that OA = A, OB = B, and OC = c express OD and OE in terms of a,b and c.

Answer 1

The question does not contain complete information that is needed to resolve it. Depending on angles $\angle BOC$ and $\angle AOB$ , the answer may be different. If we change these angles and and all conditions are done, OD and OE change. It is shown on the picture

Solution:

Introduce vectors: $\overrightarrow{OA}=\overrightarrow{a}$ , $\overrightarrow{OB}=\overrightarrow{b}$ , $\overrightarrow{OC}=\overrightarrow{c}$ .

Then $\overrightarrow{OD}=\frac{\overrightarrow{b}+\overrightarrow{c}}{2}$ (used BD = DC).

$\overrightarrow{DA}=\overrightarrow{a}-\overrightarrow{OD}=\overrightarrow{a}-\frac{\overrightarrow{b}+\overrightarrow{c}}{2}=\frac{2\overrightarrow{a}-(\overrightarrow{b}+\overrightarrow{c})}{2}$ .

AD and AM are medians. E is the point of the intersection of AD and AM. In this point each median divides as 2:1 (AE:AD = 2:1). That's why

$\overrightarrow{DE}=\frac13\overrightarrow{DA}=\frac{2\overrightarrow{a}-(\overrightarrow{b}+\overrightarrow{c})}{6}$

$\overrightarrow{OE}=\overrightarrow{OD}+\overrightarrow{DE} =\frac{\overrightarrow{b}+\overrightarrow{c}}{2}-\frac{2\overrightarrow{a}-(\overrightarrow{b}+\overrightarrow{c})}{2}=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}$

$OD=|\overrightarrow{OD}|=|\frac{\overrightarrow{b}+\overrightarrow{c}}{2}|=\frac12\sqrt{(\overrightarrow{b}+\overrightarrow{c})^2}=\frac12\sqrt{b^2+c^2+2bc\cos{\angle{BOC}}}$

$OE=|\overrightarrow{OE}|=|\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}|=\frac13\sqrt{(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})^2}=$

$=\frac13\sqrt{a^2+b^2+c^2+2(ab\cos{\angle{AOB}}+bc\cos{\angle{BOC}}+ac\cos{\angle{AOC}})}$

An answer depends on angles $\angle BOC$ , $\angle AOB$ and $\angle AOC = \angle BOC+ \angle AOB$.

We are free to change them without breaking any condition (as shown on the picture). So these angles cannot be expressed in terms of a, b and c.