Geometry: An Interactive Journey to Mastery

Lesson 35: Complex Numbers in Geometry

Example 1

The median of a triangle is a line from one vertex of a

triangle to the midpoint of its opposite side. Each triangle

has three medians.

Use complex numbers to prove that the medians of a triangle

DUHVXUHWREHFRQFXUUHQWͼ௘WKDWLVDUHVXUHWRSDVVWKURXJKD

FRPPRQSRLQW௘ͽ$OVRVKRZWKDWWKHFRPPRQSRLQWRI

intersection is^13 DORQJHDFKPHGLDQͼ௘6HHFigure 35.1௘ͽ

Solution

&RQVLGHUDWULDQJOHABC with A, B, and C regarded as complex numbers. Then, the midpoints of its sides are

MNO BC 222 ,,.C A AB

Let P be the point^13 along the median MA. We have

PM MAM     33333233112121 JJJK AM M A ©¹ ̈ ̧§·BC AABC.

Let Q be the point^13 along the median NB. We have

QN NBN     33 33323311 212 1JJJK BN N B ©¹ ̈ ̧§·AC BABC.

Let R be the point^13 along the median OC. We have

RO OCO     33333233 11212 1JJJK CO O C ©¹ ̈ ̧§·AB CABC.

These are all the same point!

Study Tip

x This lesson is purely optional. There are no recommended study tips for this lesson other than to enjoy

the lesson and let the thinking about it strengthen your understanding of geometry as a whole.

Pitfall

x The mathematics in this lesson is visually overwhelming—but it is not conceptually overwhelming.

Just be steady in your resolve as you work your way through the algebra.

A

O

B

M

C

N

Figure 35.1