Answer to Question #158467 in Geometry for Kelvin Sowah

Question #158467

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.


1
Expert's answer
2021-02-01T12:00:23-0500

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}\n=\\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.


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