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Lecture 3: Important Theorems and Properties of Triangles

Jul 16, 2024

Lecture 3: Important Theorems and Properties of Triangles

Introduction

  • Presenter: Ravi Prakash
  • Focus: Important points related to triangles
    • Properties of medians
    • Apollonius' theorem
    • Angle bisector theorem

Apollonius' Theorem

  • Definition: Relates to the medians of a triangle.
  • Formula: For a triangle ABC with medians AD, BE, CF: 2 * AD² + CD² = AB² + AC² 2 * BE² + AE² = AB² + BC² 2 * CF² + BF² = AC² + BC²
  • Application:
    • Use when finding the length of medians or related calculations.

Conclusions from Apollonius' Theorem

  1. Combination of Equations:
    • Adding three derived equations results in: 3 * (AB² + BC² + CA²) = 4 * (AD² + BE² + CF²)
  2. Centroid (G) Properties:
    • Relationship involving centroid: (AG² + BG² + CG²) / (AB² + BC² + CA²) = 1/3
  3. Perimeter Relation:
    • The perimeter of a triangle is always less than 4/3 times the sum of medians but greater than the sum of the medians.
    Perimeter < 4/3 * Sum of medians > Sum of medians

Area of the Triangle Using Medians

  • Finding Area:

    • If the lengths of the medians are given, the area of the triangle formed by these medians is 3/4 of the area of the original triangle.

    • Example: For medians 9, 12, and 15:

      Imaginary triangle area = 1/2 * 9 * 12 = 54 Original Triangle Area = 54 * 4/3 = 72

Distance Between Circumcenter and Incenter

  • Formula: Given by d² = R * (R - 2r)
    • R = Circumradius
    • r = Inradius
  • Properties:
    • Circumradius to inradius ratio R/r ≥ 2
    • Equality holds only for equilateral triangles.

Methods to Find the Area of a Triangle

  1. Heron's Formula:

    Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2

  2. Base & Height:

    Area = 1/2 * base * height

  3. Using Inradius:

    Area = r * s

  4. Using Circumradius:

    Area = (ABC) / (4R)

  5. Using Sine Rule:

    Area = 1/2 * a * b * sin(θ)

Conclusion

  • Next steps: Practice questions based on these theorems and their applications.
  • Upcoming topics: More important points about triangles and their properties.
  • Thank you for attending the lecture!

Full transcript