Triangle Geometry: Angle Bisectors, Incenter, Median

Triangles contain many useful properties related to angle bisectors, medians, altitudes and triangle centers. These results frequently appear in competitive exams like SSC, Banking and Railway exams.

1. Angle Bisector Theorem

If a line from vertex A bisects angle A of triangle ABC and meets BC at D, then:$$\frac{AB}{AC} = \frac{BD}{DC}$$ACAB​=DCBD​

This means the opposite side is divided in the ratio of the adjacent sides.

If$$AB:AC = 3:2$$AB:AC=3:2

then$$BD:DC = 3:2$$BD:DC=3:2

So BD = 3k and DC = 2k.

2. Exterior Angle Bisector

If the bisector of external angle A meets BC extended at D, then:$$\frac{AB}{AC} = \frac{BD}{DC}$$ACAB​=DCBD​

But BD and DC lie outside the triangle.

Exam Trick

Internal bisector → divides side internally

External bisector → divides side externally

3. Orthocenter of Triangle

The orthocenter (O) is the intersection point of the three altitudes of a triangle.

Important Property

$$\angle AOC = 180^\circ – \angle B$$∠AOC=180∘−∠B

This result is very common in geometry problems.

4. Incenter of Triangle

The incenter (I) is the intersection of the three internal angle bisectors.

The incenter is the center of the incircle.

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Inradius Formula

$$r = \frac{\Delta}{s}$$r=sΔ​

where

• $r$r = inradius
• $\Delta$Δ = area of triangle
• $s$s = semiperimeter

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Semiperimeter

$$s = \frac{a + b + c}{2}$$s=2a+b+c​

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Side Relations

$$x + y + z = s$$x+y+z=s $$x = s – a$$x=s−a $$y = s – b$$y=s−b $$z = s – c$$z=s−c

5. Length of Angle Bisector

The length of the angle bisector from A is:$$AD = \frac{2AB \cdot AC \cdot \cos(A/2)}{AB + AC}$$AD=AB+AC2AB⋅AC⋅cos(A/2)​

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If AB = AC (isosceles triangle)

Then

Angle bisector = median = altitude

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6. Median Theorem

If AD is the median of triangle ABC, then:$$AB^2 + AC^2 = 2(AD^2 + BD^2)$$AB2+AC2=2(AD2+BD2)

Since D is midpoint$$BD = DC$$BD=DC

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Median Diagram

7. Centroid Property

The centroid (G) is the intersection of the three medians.

It divides each median in the ratio:$$AG : GD = 2 : 1$$AG:GD=2:1

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Exam Trick

Centroid always lies inside the triangle.

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8. Angle Relation in Triangle

If M and N lie on BC, then$$\angle MAN = \frac{1}{2}(\angle B – \angle C)$$∠MAN=21​(∠B−∠C)

This result appears in advanced geometry problems.

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9. Right Triangle Projection Theorem

If altitude from A meets BC at D:$$AB^2 = BD \times BC$$AB2=BD×BC $$AC^2 = CD \times BC$$AC2=CD×BC

10. Equilateral Triangle Formulas

If side = a

Height

$$h = \frac{\sqrt{3}}{2}a$$h=23​​a

Inradius

$$r = \frac{a}{2\sqrt{3}}$$r=23​a​

Circumradius

$$R = \frac{a}{\sqrt{3}}$$R=3​a​

11. Right Triangle Radius Relations

For a right triangle:

Circumradius

$$R = \frac{\text{Hypotenuse}}{2}$$R=2Hypotenuse​

Inradius

$$r = \frac{a + b – c}{2}$$r=2a+b−c​

where

• $a$a = base
• $b$b = perpendicular
• $c$c = hypotenuse