in the isosceles triangle ABC (AB = AC), M and N are assigned respectively to the sides AB and AC such that AM = AN. Prove that the triangles BCM and BCN are congruent

Question

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Answer 1

Since AM=AN, triangle AMN is also isosceles. Angle A is common to triangles ABC and AMN. Because ABC is also isosceles and they share angle A, AMN is similar to ABC. Therefore, MN is parallel to BC. Angles AMN, ABC, ANM, ACB have equal measure because both triangles are isosceles.

AB=AM+MB, AC=AN+NC, But AB=AC and AM=AN, therefore MB=NC. BC is common to triangles BCM and BCN.

So the triangles are congruent (two sides and the included angle): SAS－MB=NC, angles MBC=ABC and NCB=ACB have the same measure, common side BC.