Geometry Class 2: Key Concepts of Triangles and Circles

Instructor: Ravi Prakash

Introduction

• Focus on CAT and ZAT exam questions
• Main topics: triangles and circles

Triangles

1. Main Concepts
   • Geometric Centers
   • Start with triangles, then move to circles

Geometric Centers of Triangles

1. Centroid

   • Intersection point of all medians
   • Definition of Median: Line joining the midpoint of a side to the opposite vertex
   • Properties:
     • Divides medians in the ratio 2:1 (from vertex to centroid)
     • Divides the triangle into 6 triangles of equal area

2. Orthocenter

   • Intersection point of all altitudes (heights)
   • Definition of Altitude: Perpendicular line from a vertex to the opposite side
   • Property:
     • Angle BOC + angle A = 180 degrees (Applicable for all vertices)

3. Incenter

   • Intersection point of all angle bisectors
   • Definition of Angle Bisector: Line dividing an angle into two equal parts
   • Properties:
     • Angle BIC = 90 + ½A, Angle AIB = 90 + ½C, Angle AIC = 90 + ½B
     • Forms an incircle touching all three sides, radius called the inradius (r)

4. Circumcenter

   • Intersection point of all perpendicular side bisectors
   • Definition of Perpendicular Bisector: Line perpendicular to a side and passing through its midpoint
   • Properties:
     • Equidistant from all three vertices, forming a circumcircle, radius called the circumradius (R)
     • Angle formed at center (e.g., BOC) is double the angle at the circumference (2θ if θ at vertex)

Important Theorems and Proofs

• Proofs provided help solidify understanding but are not always necessary for exams
• Cat/Zat exams focus on application of concepts

Summarized Properties

1. Centroid:
   • Divides medians in ratio 2:1
   • Creates 6 equal area triangles within the original triangle
2. Orthocenter:
   • Sum of angle at orthocenter and opposite vertex equals 180°
3. Incenter:
   • Angles with incenter and opposite vertex sum to 90° plus half of the included angle
   • Forms incircle that touches all three sides
4. Circumcenter:
   • Creates circumcircle touching all 3 vertices
   • Angle at center is twice the angle at the vertex formed by the same side
   • Equidistant to all vertices

Additional Points

• Root Words: Helpful in understanding – 'Ortho' means straight, 'In' means inside, 'Circum' means outside
• Circle Geometry:
  • Radius and tangent of a circle always form a 90° angle (applicable in inscribed and circumscribed circles)
  • Chord properties: Angle subtended at the center is twice the angle subtended at the circumference

Conclusion

• Focus on applications rather than rigorous proofs for exams
• Next session will discuss theorems and their applications in geometry