Lecture Notes on Geometry Concepts

Introduction

• A welcome note to students for a revision session covering important geometry concepts.

Internal Section Formula

• For points P and Q with coordinates (x1, y1) and (x2, y2), dividing a line segment in the ratio m:n:
  • X-coordinate:
    [ P_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} ]
  • Y-coordinate:
    [ P_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} ]
• External Section Formula:
  • Changes to have a minus sign:
    [ P_x = \frac{m \cdot x_2 - n \cdot x_1}{m - n} ]
  • [ P_y = \frac{m \cdot y_2 - n \cdot y_1}{m - n} ]

Triangle Centers

Centroid (G)

• Intersection point of medians of the triangle.
• Coordinates:
  [ G_{x} = \frac{x_1 + x_2 + x_3}{3} ]
  [ G_{y} = \frac{y_1 + y_2 + y_3}{3} ]
• Property:
  • Divides each median in a 2:1 ratio.
• Midpoint Triangle DEF Area:
  • Area of DEF is ( \frac{1}{4} ) of triangle ABC.

Incenter (I)

• Intersection point of angle bisectors.
• Coordinates:
  [ I_{x} = \frac{a \cdot x_1 + b \cdot x_2 + c \cdot x_3}{a + b + c} ]
  [ I_{y} = \frac{a \cdot y_1 + b \cdot y_2 + c \cdot y_3}{a + b + c} ]
• Property:
  • Incenter divides the opposite side in proportion to the adjacent sides.

Orthocenter (H)

• The point where altitudes intersect.
• Right Triangle:
  • Vertically at right angle vertex.
• Obtuse Triangle:
  • Outside the triangle.

Circumcenter (O)

• Intersection point of perpendicular bisectors.
• Properties:
  • Circumcenter is equidistant from all vertices.
  • For obtuse triangles, circumcenter is outside.

Area of Triangle

• For vertices ((x_1, y_1), (x_2, y_2), (x_3, y_3):)
  [ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| ]

Locus

• Definition: The path traced by a moving point under specific constraints.
• Example: A point moving at a unit distance from the origin describes a circle.

Shifting of Origin

• New coordinates after shifting:
  [ x' = x - a ]
  [ y' = y - b ]

Line Equations

1. Two-point Form: (y - y_1 = m(x - x_1))
2. Slope-Intercept Form: (y = mx + c)
3. Intercept Form:
   [ \frac{x}{a} + \frac{y}{b} = 1 ]
4. Parametric Form: A point and direction defined. [ x = x_0 + r \cos(\theta) ]
   [ y = y_0 + r \sin(\theta) ]

Important Properties

• Parallel Lines: Have the same slope.
• Perpendicular Lines: Slope product equals -1.

Distance from a Point to a Line

• Formula:
  [ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} ]

Foot of the Perpendicular

• Formula for reflection & calculations detailed.

Joint Equation

• For two lines given, the joint equation would be the product of the two line equations.

Homogeneous Equations

• Represent pairs of lines if delta (determinant) equals zero.

Final Remarks

• Important formula and properties summarized for quick reference.

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Remember:

• Equidistant properties for angle bisectors.
• Use properties to find intersections and distances in triangles efficiently!