New Solution of Plemelj's Triangle Problem
New Solution of Plemelj's Triangle Problem
The problem is to construct a triangle given the length of its base, the length of the altitude from to and the difference of the angles at and . This Demonstration shows Gerd Baron's solution of this construction problem using the Apollonius circle.
ABC
AB=c
h
C
C
AB
δ
A
B
Construction
Step 1: On line , choose a point . Draw a vertical segment of length . From , draw a ray at angle with respect to the segment . Let be the intersection of the ray and . From , draw a ray at angle and opposite to with respect to the segment . Let be the intersection of the ray and . The triangle is right angled. Let be the midpoint of .
λ
N
NC
h
C
C
ρ
δ/2
CN
F
ρ
λ
C
τ
π/2-δ/2
ρ
CN
E
τ
λ
EFC
S
EF
Step 2: Draw a circumcircle of the triangle with center . This circle is the Apollonius circle of the triangle . Choose any point on the circle and draw a tangent of length with as the midpoint of .
κ
EFC
S
ABC
T
PQ
c
T
PQ
Step 3: On , draw a point such that .
λ
M
SM=SQ
Step 4: On , draw the point so that . On , measure out the point so that .
λ
A
AM=c/2
τ
B
MB=c/2
Step 5: The triangle meets the stated conditions.
ABC
Verification
The correctness of the construction follows from properties of the Apollonius circle of the triangle . The radius of the Apollonius circle is /cosδ and is independent of .
ABC
h
C
c