Question 1177132
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given: triangle ABC is scalene
       altitudes AE, CD

prove: AE is not congruent to CD
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<pre>
The fact that triangle ABC is a scalene means that there is no congruent pair of sides.


The area of triangle ABC is

    area = (1/2)*|BC|*|AE| = (1/2)*|AB|*|CD|.    (1)


Assume for a minute that AE is congruent to CD.

Then from equality (1), by canceling equal factors (1/2), |AE| and |CD|, we get

    |BC| = |AB|.


But it contradicts to the given fact that triangle ABC is scalene.


The contradiction proves that AE is not congruent to CD.
</pre>

Solved.