Hello and welcome to the third post of geometry concept series and in this article we will learn about the theorems of triangles. So let's start with the first one that is Pythagoras Theorem.

According to this theorem, the square of the hypotenuse of a right angled triangle is equal to the sum of the sum of the squares of the other two sides

H² = B² + P²

AC² = AB² + BC². Its converse is also true.

The numbers which satisfy ths relation are called Pythagorean Triplets.

For eg. (5, 4, 3), (13, 12, 5), (25, 24, 7), (17, 15, 8), (41, 40, 9), (61, 60, 11), (37, 35, 12), (65, 63, 16), (29, 21, 20), (53, 45, 28), (65, 56, 33)

H² = B² + P²

If the angles of the triangle are 45°, 45° and 90°, then the hypotenuse is √2 times of any smaller side.

Excluding the hypotenuse rest of the two sides are equal.

That is....

AB = BC

AC = √2AB

AC = √2BC

If the angles of a triangle are 30°, 60° and 90° then the sides opposite to 30° angle is half of the hypotenuse and the side opposite to 60° is √3⁄2 times the hypotenuse e.g., AB = AC/2 and BC = √3⁄2AC

Therefore AB:BC:AC = 1:√3:2

Any line parallel to one side of a triangle divides the other two sides proportionally. So, if DE is drawn parallel to BC, it would divide sides AB and AC proportionally.

That is:

If the mid-points of two adjacent sides of a triangle are joined by a line segment, then this segment is parallel to the third side, i.e., if AD = DB and AE = CE then DE | | BC.

In a triangle the sum of the squares of any two sides of a triangle is equal to twice the sum of the square of the of the median to the third side and square of half the third side i.e, AB² + AC² = 2(AD² + BD²)

In a triangle the angle bisector of an angle divides the opposite side to the angle in the ratio of the remaining two sides i.e., BD/CD = AB/BC and BD X AC - CD X AB = AD²

In a triangle the angle bisector of any exterior angle of a triangle divides the side opposite to the external angle in the ratio of the remaining two sides i.e., BE/AE = BC/AC

## 1. Pythagoras Theorem

According to this theorem, the square of the hypotenuse of a right angled triangle is equal to the sum of the sum of the squares of the other two sides

H² = B² + P²

AC² = AB² + BC². Its converse is also true.

The numbers which satisfy ths relation are called Pythagorean Triplets.

For eg. (5, 4, 3), (13, 12, 5), (25, 24, 7), (17, 15, 8), (41, 40, 9), (61, 60, 11), (37, 35, 12), (65, 63, 16), (29, 21, 20), (53, 45, 28), (65, 56, 33)

H² = B² + P²

- (5)² = (4)² + (3)²
- (13)² = (12)² + (5)²
- (25)² = (24)² + (7)²
- (17)² = (15)² + (8)²
- (41)² = (40)² + (9)²
- (61)² = (60)² + (11)²
- (37)² = (35)² + (12)²
- (65)² = (63)² + (16)²
- (29)² = (21)² + (20)²
- (53)² = (45)² + (28)²
- (65)² = (56)² + (33)²

## 2. Triangle Theorem (45° - 45° - 90°)

If the angles of the triangle are 45°, 45° and 90°, then the hypotenuse is √2 times of any smaller side.

Excluding the hypotenuse rest of the two sides are equal.

∠A = 45°, ∠C = 45° and ∠B = 90° |

AB = BC

AC = √2AB

AC = √2BC

## 3. Triangle Theorem (30° - 60° - 90°)

If the angles of a triangle are 30°, 60° and 90° then the sides opposite to 30° angle is half of the hypotenuse and the side opposite to 60° is √3⁄2 times the hypotenuse e.g., AB = AC/2 and BC = √3⁄2AC

Therefore AB:BC:AC = 1:√3:2

## 4. Basic Proportionality Theorem (BPT) / Thales Theorem

Any line parallel to one side of a triangle divides the other two sides proportionally. So, if DE is drawn parallel to BC, it would divide sides AB and AC proportionally.

That is:

## 5. Mid-Point Theorem

If the mid-points of two adjacent sides of a triangle are joined by a line segment, then this segment is parallel to the third side, i.e., if AD = DB and AE = CE then DE | | BC.

## 6. Apollonius Theorem

In a triangle the sum of the squares of any two sides of a triangle is equal to twice the sum of the square of the of the median to the third side and square of half the third side i.e, AB² + AC² = 2(AD² + BD²)

BD = CD and AD is the median |

## 7. Interior Angle Bisector Theorem

In a triangle the angle bisector of an angle divides the opposite side to the angle in the ratio of the remaining two sides i.e., BD/CD = AB/BC and BD X AC - CD X AB = AD²

## 8. Exterior Angle Bisector Theorem

In a triangle the angle bisector of any exterior angle of a triangle divides the side opposite to the external angle in the ratio of the remaining two sides i.e., BE/AE = BC/AC

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