Geometry: Concepts of Triangles and Circles

Introduction

• Instructor: Ray Prakash
• Focus: Triangles and Circles
• Relevant to CAT exam questions

Geometric Centers in Triangles

1. Centroid

• Definition: Intersection point of all the medians.
• Median: Line joining a vertex to the midpoint of the opposite side.
  • Example: Triangle ABC with D as the midpoint of BC; AD is a median.
• Properties:
  • Divides median in the ratio 2:1 from the vertex.
  • Divides triangle into six smaller triangles of equal area.

2. Orthocenter

• Definition: Intersection point of all the altitudes.
• Altitude: Perpendicular from a vertex to the opposite side.
  • Example: Triangle ABC; AD is an altitude.
• Properties:
  • The sum of angles BOC and angle A = 180°.
  • Applicable for angles formed by orthocenter with any two vertices and the opposite angle.

3. Incenter

• Definition: Intersection point of all the angle bisectors.
• Angle Bisector: Line dividing an angle into two equal parts.
  • Example: Triangle ABC with bisectors AD, BE, CF intersecting at point I.
• Properties:
  • Each angle BIC, AIB, AIC = 90° + (1/2) of the opposite angle.
  • Can form an in-circle that touches all sides of the triangle at exactly one point.
  • Equidistant from all sides of the triangle.

4. Circumcenter

• Definition: Intersection point of all the perpendicular bisectors of the sides.
• Perpendicular Bisector: Line perpendicular to a side and bisecting it.
  • Example: Perpendicular bisectors of sides of triangle ABC intersect at point O.
• Properties:
  • Can form a circumcircle passing through all three vertices of the triangle.
  • Capital R represents circumradius; the circle's radius.
  • Circumcenter is equidistant from all vertices.
  • An angle subtended by a chord at the center is double the angle subtended at any point on the circumference.

Summary

Four main geometric centers in triangles: Centroid, Orthocenter, Incenter, and Circumcenter, each with unique properties and significance in triangle geometry. Understanding these concepts is crucial for solving relevant geometry problems in competitive exams like the CAT.

Conclusion

Next class will cover applications and theorems related to these geometric centers in triangles.