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

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

This Demonstration shows how to construct a triangle given the length of the side opposite the vertex , the sum of the lengths of the other two sides and a line containing the vertex . Since lies on the ellipse with foci and , the semi-axes of the ellipse are and .

ABC

c

C

a+b

ρ

C

C

A

B

d=(a+b)/2

e=1/2-

2

(a+b)

2

c

Let be the circle with center at the midpoint of and of radius . Let the points and be on the perpendicular bisector of at heights and , respectively. Let and be the lines parallel to through and . Let . Let the line through perpendicular to meet at . Let the line intersect at . Then the point lies on the line perpendicular to through such that the ratio of the distances of and from is .

σ

AB

d

E

F

AB

e

d

λ

1

λ

2

AB

F

E

G=ρ⋂

λ

2

G

AB

λ

1

H

τ=DH

σ

I

C

AB

I

I

C

AB

d:e