Triangles 1

In Tri(ABC), if AD is the angular bisector, then, AB/AC = BD/DC.

In Tri (ABC), if E and F are points on AB and AC respectively, and EF || BC, then, AE/AB = AF/AC.

In Tri (ABC), if AD is a median, then AB^2 + AC^2 = 2*(AD^2 + BD^2).

Apollonius’s Theorem:

 

LM^2 + LN^2 = 2*(MO^2 + LO^2).

This can be applied to 4 triangles in a parallelogram.

LN^2 + LN^2 = 2*LO^2 + (1/2)*MN^2.

For triangle ABC,

Sine rule: a/(sin A) = b/(sin B) = c/(sin C).

Cosine Rule: cos A = (b^2 + c^2 – a^2)/(2*b*c).
                       cos B = (a^2 + c^2 – b^2)/(2*a*c).
                       cos C = (a^2 + b^2 – c^2)/(2*a*b).

Angular bisector theorem:  BD/CD = AB/AC.

For an acute angled triangle,

AC^2 = AB^2 + BC^2 – 2*BC*BD.

 

For an obtuse angled triangle,

AC^2 = AB^2 + BC^2 + 2*BC*BD. 

  

For a right angled triangle, when AD is perpendicular to BC,

AD^2 = BD*DC.
AB^2 = BD*BC.
AC^2 = CD*BC.