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In triangle ABC, the angle bisector of \(\angle BAC\) meets \(\overline{BC}\) at D, such that AD = AB. Line segment \( \overline{AD}\) is extended to E, such that CD = CE and\( \angle DBE = \angle BAD\). Show that triangle ACE is isosceles.

 
 Mar 18, 2020
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Note that angle CEA = angle DCA + angle DAC and angle BDE = angle DCE + angle EDC, so angle DCA = angle CEA - angle DAC and angle DCE = angle EDC - angle CED.

 

Also, angle BDE = angle ABD + angle ADB = angle AEC + angle DAC, so angle DCE + angle DCA = (angle EDC - angle CED) + angle DCA.

 

Therefore, angle ACE = angle ACD + angle DCE = angle CEA, which implies that triangle ACE is isosceles.

 Feb 7, 2024

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