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### Intersecting Vectors and the Dot Product

```
Date: 04/24/98 at 03:05:47
From: Tony Lu
Subject: Geometry (vector)

Each of the following geometrical theorems can be proved by
vectors using dot product(scalar product). Prove that:

i) the altitudes of a triangle are concurrent
ii) the perpendicular bisectors of the sides of a triangle
are concurrent

I cannot figure out a way to prove that the three vectors intersect
at a common point. I am not sure which properties of the scalar

Thanks.
```

```
Date: 04/24/98 at 08:30:00
From: Doctor Jerry
Subject: Re: Geometry(vector)

Hi Tony,

I'm not sure that what I'll suggest is the shortest, most elegant way,
but I think it is correct.

Let vertices of triangle be A, B, and C; let b be the vector from A to
C, a the vector from C to B, and c be the vector from B to A.

The altitude from A perpendicular to a can be written as:

A:  b + L_A*a = h_A

where L_A (L sub A) is a scalar and h_A is a vector stretching from A
to side BC. Similarly:

B:  c + L_B*b = h_B

C:  a + L_C*c = h_C

where h_A dot a = 0, h_B dot b = 0, and h_C dot c = 0.

First show that the line through A and parallel to h_A intersects the
line through B and parallel to h_C intersect. Use A as origin for
writing an equation for these lines:

r = t*h_A

r = b + s*h_C

where s and t are parameters. Set these equal, dot both sides with b,
and solve for s:  (I'll use @ as dot)

s = -(a@b)/(a@h_C)

Now we must show the point b+s*h_C (with the above s) is on the line

b+s*h_C = -c+w*h_B

has a solution.

Dot both sides with c.

I think this works out.  Please check my work.

-Doctor Jerry,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
College Euclidean Geometry
College Linear Algebra
High School Euclidean/Plane Geometry
High School Linear Algebra

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