Triangle Part - II ( Advanced)

If two triangles are congruent they have equal sides, equal areas. 

Condition for Congruence

SAS Theorem:
If two sides and the angle between these two sides (included angle) of one triangle are congruent to the corresponding two sides and the included angle of a saecond triangle, then the two triangles are congruent

Example:

AAS Theorem:
If two angles and a side not between these two angles of one triangle are congruent to two angles and the corresponding side not between these two angles of a second triangle, then the two triangles are congruent

Example:

Hypotenuse-Leg Theorem:

If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and corresponding leg of a second right triangle, then the two triangles are congruent

Example:

Pythagoras’ Theorem: In a right angle triangle the square of the hypotenuse is equal to the sum of the squares of other two sides.

b²=a²+c²

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, 

 AB²+AC²=2(AD²+BD²)=2(AD²+DC²)

Stewart theorem:

In Triangle ABC, AD divides side BC in the ratio m and n. (Here AD need not be median) then, 

m.b²+n.c²=a(d²+mn)

Mean proportionality and Mid Point theorem:

In the first triangle, DE // BC so AD / DB = AE / EC

In the second triangle, D and E are mid points of AB and AC respectively. Which implies,
AD / DB = AE / EC = 1

Also, DE = 1 / 2 BC

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