Points D and E are chosen on the AC side of triangle ABC so that the segments AD and CE are equal.

Points D and E are chosen on the AC side of triangle ABC so that the segments AD and CE are equal. It turned out that the segments BD and BE are also equal. Prove that triangle ABC is isosceles.

1. Let’s draw a picture.

2. Consider the triangle DBE.

This is an isosceles triangle, since by the condition BD = BE.

∠BDE = ∠BED, as these are the angles at the base of an isosceles triangle.

3. Let us define ∠BDA and ∠BEC.

∠BDA and ∠BDE are adjacent, therefore

∠BDA = 180 ° – ∠BDE.

Similarly, ∠BEC and ∠BED are adjacent, so

∠BEC = 180 ° – ∠BED.

Since ∠BDE = ∠BED, then ∠BDA = ∠BEC.

4. Consider triangles ABD and CBE.

These triangles are equal in two sides and the angle between them:

BD = BE and AD = CE – by condition;

∠BDA = ∠BEC.

Therefore, both sides BA and BC are equal.

Hence, triangle ABC is isosceles.