The Plemelj Construction of a Triangle: 12
The Plemelj Construction of a Triangle: 12
This Demonstration constructs a triangle given the length of its base , the length of the altitude from to , and the difference between the angles at and at . This construction is an alternative to The Plemelj Construction of a Triangle: 5.
ABC
c
AB
h
C
C
AB
δ
α
A
β
B
Construction
Draw a line segment of length and let the midpoint of be . Draw a line segment of length perpendicular to .
AB
c
AB
M
BB'
h
C
AB
Step 1: Draw a circle with center so that the chord subtends the angle , which implies the central angle .
σ
O
AB
δ
∠AOB=2δ
Step 2: Let be the intersection of the ray and the circle .
D'
MB'
σ
Step 3: Extend to so that is the midpoint of .
D'M
D''
M
D'D''
Step 4: Draw a ray parallel to at distance above . The point is the intersection of and .
ρ
AB
h
C
AB
C
D''B
ρ
Step 5: The triangle meets the stated conditions.
ABC
Verification
Let be the intersection of the segment and .
E
AD''
ρ
The quadrilateral is a parallelogram. The angle at is .
AD'BD''
D''
δ
The triangle is isosceles, so the angle . On the other hand, . So and .
EAC
∠EAC=π-2α
∠EAC=π-(π-γ)-δ
2α=(α+β)-δ
δ=α-β