In a triangle ABC with angle A

Question

(A) For two triangles to be similar,their corresponding angles must be equal. In $	riangle ABC$,the angles are ordered as $\angle A < \angle B < \ang

Vedclass · Accepted Answer

$	riangle ABD$

Vedclass · Answer

(A) For two triangles to be similar,their corresponding angles must be equal. In $	riangle ABC$,the angles are ordered as $\angle A Consider $	riangle ABD$. Since $D$ is on the interior of segment $BC$,$\angle ADB$ is an exterior angle to $	riangle ADC$,so $\angle ADB = \angle DAC + \angle C$. Thus,$\angle ADB > \angle C$. Since the largest angle of $	riangle ABC$ is $\angle C$,and $	riangle ABD$ contains an angle $\angle ADB$ which is strictly greater than $\angle C$,the set of angles in $	riangle ABD$ cannot be the same as the set of angles in $	riangle ABC$. Therefore,$	riangle ABD$ cannot be similar to $	riangle ABC$.Similarly,one can show that $	riangle BCE$ and $	riangle CAF$ cannot be similar to $	riangle ABC$ under general conditions. However,among the given options,$	riangle ABD$ is a standard example of a triangle formed by a cevian that cannot be similar to the original triangle due to the angle inequality constraints.

In a $\triangle ABC$ with $\angle A < \angle B < \angle C$,points $D, E, F$ are on the interior of segments $BC, CA, AB$ respectively. Which of the following triangles cannot be similar to $\triangle ABC$?

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