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

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

This Demonstration shows the construction of a triangle given the length of the base , the ratio of the other two sides and a line containing .

ABC

c

AB

λ=b/a

μ

C

Construction

1. Draw a line and a line through and .

μ

σ

A

B

2. Draw the point such that . Draw the points and so that and is parallel to . Let be the point where and intersect. Let be the point where meets .

D

BD=a

H

I

AH=AI=b

HI

BD

F

DI

σ

E

DH

σ

3. Let be the midpoint of . Draw the circle with center and radius .

S

EF

κ

S

r=SF

4. The point is an intersection of and .

C

κ

μ

Verification

The circle is the Apollonius circle of the points and with respect to the ratio .

κ

A

B

λ=b/a