28. Construct a Triangle ABC Given the Length of AB, the Sum of the Other Two Sides, and a Line Containing C

c

1

a + b

1.4

angle of line ρ (in radians)

ϕ

0.7

x intercept of ρ, so that D = (x, 0)

x

-0.1

zoom

1

show ellipse

This Demonstration shows how to construct a triangle

ABC

given the length

c

of the side opposite the vertex

C

, the sum of the lengths of the other two sides

a+b

and a line

ρ

containing the vertex

C

. Since

C

lies on the ellipse with foci

A

and

B

, the semi-axes of the ellipse are

d=(a+b)/2

and

e=1/2

2

(a+b)

-

2

c

.

Let

σ

be the circle with center at the midpoint of

AB

and of radius

d

. Let the points

E

and

F

be on the perpendicular bisector of

AB

at heights

e

and

d

, respectively. Let

λ

1

and

λ

2

be the lines parallel to

AB

through

F

and

E

. Let

G=ρ⋂

λ

2

. Let the line through

G

perpendicular to

AB

meet

λ

1

at

H

. Let the line

τ=DH

intersect

σ

at

I

. Then the point

C

lies on the line perpendicular to

AB

through

I

such that the ratio of the distances of

I

and

C

from

AB

is

d:e

.