The three angle bisectors of a triangle are concurrent at a single point. This is proved by first constructing the angle bisectors of any triangle. It is then shown through a series of steps using properties of similar triangles and perpendicular lines that the three angle bisectors intersect at the same point. Specifically, it is shown that two pairs of triangles are similar, meaning the corresponding parts are proportional. This implies that two of the angle bisectors are the same length. It is also shown that the triangles formed at one end of an angle bisector are right triangles. Putting this all together, it is proved that the angle measure of the angles where the bisectors meet the triangle sides are equal, meaning the bisectors intersect at a

1. Three Angle Bisectors
of a Triangle are Concurrent
Created by Laurado Rindira Sabatini

2. Prove that three angle bisectors of the internal angles
of a triangle are concurrent
The three angle
bisectors; AD, BE,
and FG are
concurrent because
the point C is on all
of the angle
bisectors.

3. 

Given ∆ ABC
Construct the
angle bisectors of
⦟ABC and ⦟ ACB
which intersect at
the point O

4. 
Construct
perpendicular line from
O to AB, from O to
AC, and from O to BC
No
Proof
Reason
1
∆ DOB ≈ ∆ FOB
Angle-Side-Angle
2
OD = OF
Result of step 1
3
∆ EOC ≈ ∆ FOC
Angle-Side-Angle
4
OE = OF
Result of step 3
5
OD = OE = OF
Steps 2 and 4

5. 
Construct segment AO

6. No
Proof
Reason
6
∆ ADO is right triangle
AB ⊥ OD
7
∆ AEO is right triangle
AC ⊥ OE
8
OD = OE
Step 5
9
AO = AO (The hypotenuse of ∆
ADO and ∆ AEO respectively)
Reflective property
10
∆ ADO ≈ ∆ AEO
Step 8 and 9
11
m ∠ DAO = m ∠ EAO
Step 10
12
AO is the angle bisector
Step 11
13
O lies on the angle bisector of AO,
BO, and CO
Constructed and step 12
14
The three angle bisectors of a
triangle are concurrent
Step 13
Proved

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