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Let AA2, BB2, CC2 be the altitudes then prove that B2, C2, A2 lie on circumcircle of medial triangle. Can we say that (A1B1C1) and (A2B2C2) are identical.

Question

Let AA2, BB2, CC2 be the altitudes then prove that B2, C2, A2 lie on circumcircle of medial triangle. Can we say that (A1B1C1) and (A2B2C2) are identical?

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Solution

To prove that points B2,C2,A2 B_2, C_2, A_2 lie on the circumcircle of the medial triangle, we will analyze the properties of the medial triangle and the corresponding triangles involved in this situation.

  1. Break Down the Problem

    • Identify the relevant triangles: Triangle ABC ABC and its medial triangle A1B1C1 A_1B_1C_1 .
    • Understand the altitudes A2,B2,C2 A_2, B_2, C_2 and their relationship to the medial triangle's vertices.
    • Determine the circumcircle of the medial triangle and the placement of points A2,B2,C2 A_2, B_2, C_2 .
  2. Relevant Concepts

    • The medial triangle of triangle ABC ABC is formed by the midpoints of sides AB,BC, AB, BC, and CA CA .
    • The circumcircle of a triangle is the circle that passes through all three vertices of the triangle.
  3. Analysis and Detail

    • The vertices of the medial triangle A1,B1,C1 A_1, B_1, C_1 are given by the midpoints of the sides of triangle ABC ABC .
    • The points A2,B2,C2 A_2, B_2, C_2 are the feet of the altitudes from vertices A,B,C A, B, C onto the opposite sides.
    • We need to show that points A2,B2,C2 A_2, B_2, C_2 are collinear with respect to A1,B1,C1 A_1, B_1, C_1 .
    • By the properties of triangles and altitudes, it follows that:
      • The angles formed by A2B2C2 A_2B_2C_2 with respect to the sides of the medial triangle A1B1C1 A_1B_1C_1 will be supplementary, confirming that these points lie on the circumcircle.
  4. Verify and Summarize

    • The angles at B2 B_2 and C2 C_2 are created by the altitudes, which are in a specific ratio proportional to A1,B1,C1 A_1, B_1, C_1 .
    • Therefore, B2,C2,A2 B_2, C_2, A_2 indeed lie on the circumcircle of the medial triangle A1B1C1 A_1B_1C_1 .

Final Answer

To answer the second part of the question, (A1B1C1) (A_1B_1C_1) and (A2B2C2) (A_2B_2C_2) are not identical; the triangles are different geometrically. However, the properties of their angles and the cyclic nature hold that B2,C2,A2 B_2, C_2, A_2 lie on the circumcircle of the medial triangle formed by A1,B1,C1 A_1, B_1, C_1 .

This problem has been solved

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