32b. Construct a Triangle ABC Given the Length of AB, the Ratio of the Other Two Sides and a Line through C

move B

1.3

λ

a

1.2

b

0.66

line with point C

y intercept

1

slope

0.7

change D

1.4

steps

1

2

3

4

plot range

1.4

This Demonstration shows the construction of a triangle

ABC

given the length

c

of the base

AB

, the ratio

λ=b/a

of the other two sides and a line

μ

containing

C

.

Construction

1. Draw a line

μ

and a line

σ

through

A

and

B

.

2. Draw the point

D

such that

BD=a

. Draw the points

H

and

I

so that

AH=AI=b

and

HI

is parallel to

BD

. Let

F

be the point where

DI

and

σ

intersect. Let

E

be the point where

DH

meets

σ

.

3. Let

S

be the midpoint of

EF

. Draw the circle

κ

with center

S

and radius

r=SF

.

4. The point

C

is an intersection of

κ

and

μ

.

Verification

The circle

κ

is the Apollonius circle of the points

A

and

B

with respect to the ratio

λ=b/a

.