Loci and Concurrency Theorems Tutorial

In this tutorial, we will study about definition of loci and concurrency, loci and concurrency theorems.

Definition of Loci:

If a point moves in some way which satisfies some given geometrical condition at every instant during its motion, then the path traced out by the moving point is called loci of a point.

Every point satisfies the given geometrical condition is a point of loci, and

Every point of loci should satisfy the given geometrical condition.

Singular form of loci is called as locus.

Definition of concurrency:

If three or more than three lines pass through a point, then the common point is called point of concurrency.

Let us see about loci and concurrency theorems in this tutorial.
Tutorial - Loci Theorem:

Definition:

The loci of a point which is equidistant from the two given points, is the perpendicular bisector of the line segment and joining the two given points.

Given:

There are two given fixed points M and N.

A is a moving point that is A is loci of two given points M and N.

AM = AN.

To Prove:

The loci of A is the perpendicular bisector of MN line segment.

Construction:

Triangle-Loci Theorem

Join MN. Bisect MN at B. Join AM, AN and AB.

Proof:

Therefore, A moves such that AM = AN.

As we know that,

A must pass through the point B of the line segment MN, because BM = BN.

Thus A passes through the mid-point of MN.

Now, in ∆MAB and ∆NAB,

Given    AM = AN

By construction, BM = BN

AB = AB

By sss congruence rule,

∆MAB = ∆NAB

angle MBA = angle NBA       (1)

But angle MBA + angle NBA = 180°  (2)

Substitute (1) into (2),

2 angle MBA = 180°

Divide by 2 each side.

angle MBA = 90°

Therefore, AB _|_ MN

Thus, AB is proved as perpendicular bisector of MN.
Tutorial - Concurrency Theorem:

Definition:

A concurrency point of the perpendicular bisectors of sides of a triangle is called circum-center of the triangle.

A concurrency point of the median of a triangle is called centroid of the triangle.

The concurrency diagram is shown in triangle figure beliow.

Triangle-Concurrency Theorem

A,B,C are mid-points of PQ, QR and QR respectively. PB, QC and RA are medians of ∆PQR. Here O is Concurrency point.

As we know from congruence rule,

OB _|_ QR

OA _|_ PQ

OC _|_ PR

These are all perpendicular bisectors of triangle. Therefore, O is concurrency point.