Important Concepts of Triangles.

Triangles

Centroid [Intersecting Point of medians of the triangle] : 

• Centroid divides median of a triangle in ratio 2 : 1

→  AG/GE=2/1

→ AG/AE=2/3

→ GE/AE=1/3

• area ∆ ABE =  1/2 area ∆ ABC
• area ∆ AGB = 1/3 area ∆ ABC
• area ∆ AGF = 1/6 area ∆ ABC

• AO = OD
• OG = 1/3 AO
• AB² + BC² = 2BD² + 1/2 AC²
• AB² + AC² = 2AE² + 1/2  BC²
• CA² + CB² = 1/2 AB² + 2 FC²
• Area ∆ FGE = 1/12 area ∆ ABC
• 3 × (sum of side square) = 4x (sum of median square) 
• 3 × (AB² + BC² + AC²) = 4x (AE² + BD² + CF²)
• area ∆ ABC = 4/3 area ∆ (Formed by taking AD, BF, CE, as sides of a triangle).

• Incenter- Intersecting Point of Internal angle Bisector.

• Circumcenter- Intersecting Point of Perpendicular Bisector.

• Orthocenter- Intersecting Point of Altitudes.

Important Points: -

(a) Orthocenter of right angled triangle = at right angles vertex.

(b) Circumcenter of right angled triangle = Mid-point of Hypotenuse

(c) Distance b/w incenter & circumcenter of a triangle= sq root(R^2 - 2Rr)

                                                                                (R=circumradius   r= incente)

(d) In Equilateral triangle,   R=2r

                       Circum Radius : Inradius

                                    2          :     1