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Important Theorems Related to Triangle

45° – 45° – 90°
Description: P-330-1.tif
∠A = 45° ∠B = 90° ∠C = 45°

Explanation: If the angles of a triangle are 45°, 45° and 90°, then the hypotenuse (i.e., longest side) is Description: 6768.png times of any smaller side. Excluding hypotenuse rest two sides are equal. i.e., AB = BC and AC Description: 6761.pngAB Description: 6754.png BC
AB : BC : AC = 1 : 1 : Description: 27974.png

30° – 60° – 90°
Description: P-330-2.tif
∠C = 30°, ∠B = 90°, ∠A = 60°

Explanation: 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 Description: 6739.pngtimes the hypotenuse, e.g., 
AB Description: 6732.png and Description: 6725.png AC
AB : BC: AC = 1: Description: 6718.png

Basic Proportionality Theorem (BPT)
Description: P-330-3.tif
Explanation: 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 i.e Description: 11251.png or Description: 11244.pngDescription: 11237.png.

Mid-point Theorem
Description: P-330-4.tif
Explanation: Any line joining 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 = BD and AE = CE then DE||BC.

Apollonius’ Theorem
Description: P-330-5.tif

Explanation: 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 median to the third side and square of half the third side. i.e., AB2 + AD2 = 2 (AC2 + BC2)

Stewarts Theorem/Generalization of Apollonius Theorem
Description: P-330-6.tif

Explanation: If length of AP = m and PB = n, then m × CB2 + n × AC2 = (m + n) PC2 + mn (m + n Also understand that m and n here are length of segments, and not their ratio.

Extension of Apollonius’ Theorem
Description: 23132.png

Explanation: In the given ∆ ABC, AC, BE and DF are medians. 3 (Sum of squares of sides) = 4 (Sum of squares of medians) 3 (AB2 + AD2 + DB2) = 4 (AC2 + EB2 + FD2)

Interior Angle Bisector Theorem
Description: 23147.png
Explanation: 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., Description: 6628.png and BD × AC − CD × AB = AD2

Exterior Angle Bisector Theorem
Description: 23176.png
Explanation: 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., Description: 6613.png

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