The following four theorems are most important in solving questions on triangles.
Pythagoras’ theorem:
In a right angle triangle the square of the hypotenuse is equal to the sum of the squares of other two sides.
Pythagorean triplet: There are certain triplets which satisfy the pythagoras’ theorem and are common, called pythagorean triplet. For example: 3, 4, 5; 5, 12, 13; 10, 24, 26; 7, 24, 25; 8, 15, 17
Appolonious theorem:

$\therefore$ $A{B^2} + A{C^2} = 2(A{D^2} + B{D^2}) = 2(A{D^2} + D{C^2})$
Stewart theorem:

$\therefore m.{b^2} + n.{c^2} = a({d^2} + mn)$
Mean proportionality and Mid Point theorem:

In the second triangle, D and E are mid points of AB and AC respectively.
Which implies, $\displaystyle\frac{{AD}}{{DB}} = \frac{{AE}}{{EC}} = 1$
Also, $DE = \displaystyle\frac{1}{2}BC$