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Problem Of The Week # 295 - Jan 04, 2018

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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,198
Happy New Year! Here is this year's first University-level POTW:

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Tetrahedron $OABC$ is such that lines $OA, OB,$ and $OC$ are mutually perpendicular. Prove that triangle $ABC$ is not a right-angled triangle.

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,198
Congratulations to Opalg and castor28 for their correct solutions to this week's POTW, which was Problem 63 in the MAA Challenges. castor28 's solution follows:

We can take an orthogonal coordinate system with $O$ as origin and one axis through each of $A, B,$ and $C$. The coordinates of the vertices are $A = (a, 0, 0), \; B=(0,b,0),$ and $C=(0,0,c)$.

We compute the dot product of the vectors $AB$ and $AC$:
$$ AB\cdot AC = (-a,b,0)\cdot(-a,0,c) = a^2\ne 0.$$

This shows that $\angle BAC$ is not a right angle; the same argument applies to the other two angles of the triangle $ABC$.
 
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