The Plemelj Construction of a Triangle: 9
The Plemelj Construction of a Triangle: 9
This Demonstration constructs a triangle given the length of the base , the length of the altitude from to and the difference of the angles at and . This construction unifies two constructions mentioned in The Plemelj Construction of a Triangle: 1.
ABC
c
AB
h
C
C
AB
α-β
A
B
Construction
Step 1: Draw a line segment of length and a perpendicular line segment of length with midpoint .
AB
c
BB'
h
C
H
Step 2: Draw a circle with center such that subtends an angle from points on above the chord . The angle equals .
σ
S
AB'
π/2-δ
σ
AB'
SB'A
δ
Step 3: Find a point on at distance from and a point on at distance from .
K
σ
c
A
L
σ
c
B'
Step 4: Draw the isosceles trapezoid .
B'KAL
Step 5: The point is the intersection of the straight line through parallel to and the perpendicular bisector of and .
C
H
AB
KB'
AL
Step 6: The triangle meets the stated conditions.
ABC
Verification
The three triangles , and are congruent. In the isosceles triangle , , so .
ABC
LB'C
KAC
B'KC
∠CKB'=∠LB'K-∠LB'C=π/2-δ-β
∠B'CK=2(δ+β)
On the other hand, .
∠B'CK=2π-2γ-2β=2π-2(π-α-β)-2β=2α
So and .
2(δ+β)=2α
δ=α-β
The pink, blue and green arcs have measures , and .
2β
2γ
2(δ+β)