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
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