# Midpoint

## Contents

## Definition

In Euclidean geometry, the **midpoint** of a line segment is the point on the segment equidistant from both endpoints.

A midpoint bisects the line segment that the midpoint lies on. Because of this property, we say that for any line segment with midpoint , . Alternatively, any point on such that is the midpoint of the segment.

## Midpoints and Triangles

pair A,B,C,D,E,F,G; A=(0,0); B=(4,0); C=(1,3) D=(2,0); E=(2.5,1.5); F=(0.5,1.5); G=(5/3,1); draw(A--B--C--cycle); draw(D--E--F--cycle,green); dot(A--B--C--D--E--F--G); draw(A--E,red); draw(B--F,red); draw(C--D,red); label("A",A,S); label("B",B,S); label("C",C,N); label("D",D,S); label("E",E,E); label("F",F,W); label("G",G,NE); label("Figure 2",D,4S); (Error compiling LaTeX. D=(2,0); ^ db294ad43df3fe6d80dd0c70449bc4f56aec8bee.asy: 9.1: syntax error error: could not load module 'db294ad43df3fe6d80dd0c70449bc4f56aec8bee.asy')

### Midsegments

As shown in Figure 2, is a triangle with , , midpoints on , , respectively. Connect , , (segments highlighted in green). They are midsegments to their corresponding sides. Using SAS Similarity Postulate, we can see that and likewise for and . Because of this, we know that Which is the Triangle Midsegment Theorem. Because we have a relationship between these segment lengths, with similar ratio 2:1. The area ratio is then 4:1; this tells us

### Medians

The median of a triangle is defined as one of the three line segments connecting a midpoint to its opposite vertex. As for the case of Figure 2, the medians are , , and , segments highlighted in red.

These three line segments are concurrent at point , which is otherwise known as the centroid. This concurrence can be proven through many ways, one of which involves the most simple usage of Ceva's Theorem and the properties of a midpoint. A median is always within its triangle.

The centroid is one of the points that trisect a median. For a median in any triangle, the ratio of the median's length from vertex to centroid and centroid to the base is always 2:1.

## Cartesian Plane

In the Cartesian Plane, the coordinates of the midpoint can be obtained when the two endpoints , of the line segment is known. Say that and . The Midpoint Formula states that the coordinates of can be calculated as:

## See Also

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