The Three Perpendicular Bisectors of Triangle are Concurrent

1. Created by Laurado Rindira Sabatini
Three Perpendicular Bisector
of A Triangle are Concurrent

2. Introduction
I want to show that the three perpendicular bisectors of a
triangle are concurrent.
Concurrent lines,
either segments or
rays, are lines which
lie in the same
plane and intersect
in a single point.
The point of
intersection is the
point of concurrency
Example of point of concurrency
P
Point P is called
point of concurrency

3. Proof
 Given ∆ ABC
 Construct the
perpendicular bisector
of segment AC. Label
as p
 Construct the midpoint
of segment AC. Call M

4.  Construct any point K
on the perpendicular
bisector p
 Construct segment AK
and segment KC
No
Proof
Reason
1
AM = CM
Definition of midpoint
2
KM = KM
Reflective property
3
m⦟ AMK = m ⦟ CMK Right angles
4
∆ AMK ≈ ∆ CMK
Side – Angled - Side
5
AK = CK
Step no 4

5.  Construct the
perpendicular bisector of
BC. Label as q
 Construct the midpoint of
segment BC. Call it M’
Since AC and BC are not parallel, the perpendicular bisectors,
p and q, must also intersect. Let point K be the point of intersection.

6.  Construct point K as
the point of
intersection p and q
 Construct segment BK
No
Proof
Reason
6
BM’ = CM’
Definition of midpoint
7
KM’ = KM’
Reflective property
8
m⦟ BM’K = m ⦟
CM’K
Right angles
9
∆ BM’K ≈ ∆ CM’K
Side – Angled - Side
10
BK = CK
Step no 4
Continued

7. No
Proof
Reason
11
AK = BK = CK
Step no 5 and 10
12
K is taken to be the intersection of
the perpendicular bisectors p and q
Constructed
13
K lies on the perpendicular bisector
of AB
Step 12
14
K lies on the perpendicular bisector
of AC, BC, and AB
Step 12 and 13
15
The three perpendicular bisectors of
a triangle are concurrent
Step no 14
Proved

8. Thanks for Your Attention!!!
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