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