1.6. Congruence tests for triangles, Kiselev's Geometry, Book I. Planimetry, Givental Alexander.

Exercises.

78. On each side of an equilateral triangle ABC, congruent segments AB', BC', and CA' are marked, and the points A', B', and C' are connected by lines. Prove that the triangle A'B'C' is also equilateral.

Proof: The triangle ABC is equilateral, so AB = AC and AB = BC. This implies that ∠ABC = ∠ ACB and ∠BAC = ∠BCA. So  ∠ABC = ∠BCA = ∠ACB. In other words ∠B'BC' = ∠A'AB' = ∠A'CC'. We already know that AB' = BC' = CA'. Further, AB = BC= CA implies that A'A = BB' = CC'. Thus, triangle A'CC' is congruent to C'BB' and also triangle A'CC' is congruent to A'AB', i.e. triangles A'CC', C'BB', and A'AB' are congruent to each other. We can conclude that A'C' = C'B' = A'B'. In other words, we have proved that triangle A'B'C' is equilateral.

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