In triangle ABC with obtuse angle ABC, heights AA1 and CC1 are drawn. Prove that triangles A1BC1 and ABC are similar

Consider the triangles AA1B and CC1B.
∠ABA1 = ∠CBC1, since they are vertical.
∠AA1B = ∠CC1B, since they are direct by the condition of the problem.
Therefore, according to the first sign of the similarity of triangles, these triangles are similar.
Then, by the definition of such triangles:
AB / BC = A1B / C1B
We transform this equality:
AB / A1B = BC / C1B
Consider the triangles A1BC1 and ABC.
∠ABC = ∠A1BC1, since they are vertical.
Then, according to the second sign of similarity, these triangles are similar.

Remember: The process of learning a person lasts a lifetime. The value of the same knowledge for different people may be different, it is determined by their individual characteristics and needs. Therefore, knowledge is always needed at any age and position.