Important theorems related to triangles


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.

${b^2} = {a^2} + {c^2}$

Pythagorean triplet: There are certain triplets which satisfy the pythagoras’ theorem and are commonly, called pythagorean triplet.
For example: 3, 4, 5;    5, 12, 13;    24, 10, 26;    24, 7, 25;    15, 8, 17

Appolonious theorem:  

In triangle ABC, AD is median, which divides BC into two equal parts. Then, 

 $A{B^2} + A{C^2} = 2(A{D^2} + B{D^2}) = 2(A{D^2} + D{C^2})$

Stewart theorem:

In Triangle ABC, AD divides side BC in the ratio m and n. (Here AD need not be median) then, 
$m.{b^2} + n.{c^2} = a({d^2} + mn)$

Mean proportionality and Mid Point theorem:

In the first triangle, DE // BC so $\displaystyle\frac{{AD}}{{DB}} = \frac{{AE}}{{EC}}$

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, $DE = \displaystyle\frac{1}{2}BC$

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