# Altitude and median in equilateral triangle

In equilateral triangle, the altitude and median are the same segment.

In this chapter, we will attempt to prove that altitude is also the median in equilateral triangle.

Before proving the above concept, let us review the basics.

## Basics about altitude and median

### What is equilateral triangle ?

The triangle in which all sides are of equal measure are called equilateral triangle.

Given above is the equilateral triangle in which AB = BC = CA.

In equilateral triangle, all angle measure 60 degree.

### What is altitude of triangle ?

The segment that starts from vertex and intersect the opposite side at right angle is called altitude of triangle.

In the above triangle, AM is the altitude.

Since the altitude intersect the side at right angle, \mathtt{\angle AMC\ =\ 90\ degree}

### What is median of triangle ?

The segment that start from vertex and intersect the other side in two equal halves is called median of triangle.

In the above triangle, AN is the median.

Since the median bisect the side into two equal halves, we can say that BN = NC.

## Proving altitude is also the median in equilateral triangle

Here we will try to prove that the altitude drawn in equilateral triangle is also a median.

Consider the above equilateral triangle ABC in which AM is the altitude of the triangle.

Since the triangle is equilateral, we can say that AB = BC = CA.

Taking Triangle ABM and ACM

AB = AC (sides of equilateral triangle are equal)
AM = MA (Common side)
∠AMB = ∠AMC = 90 degree

By RHS congruency, both triangles ABM and ACM are congruent.

i.e. \mathtt{\triangle ABM\ \cong \triangle ACM}

As both triangles are congruent, we can say that; BM = MC

Since BM = MC, it means that line AM is the bisector of side BC. And any line that touch vertex and bisect the other side is the median.

Hence, line AM is the median of triangle.

Conclusion
In equilateral triangle, the altitude and median are both the same line.