1. Construct a Triangle Given the Length of the Base, the Difference of the Base Angles and the Foot of the Altitude to the Base
1. Construct a Triangle Given the Length of the Base, the Difference of the Base Angles and the Foot of the Altitude to the Base
This Demonstration constructs a triangle given the length of its base, the difference of the base angles and the foot of the altitude from to the base.
AB=c
δ
D
C
Construction
Step 1: Draw a horizontal line of length and the foot of the altitude from . Measure out a point so that , so that is the midpoint of .
AB
c
D
C
E
ED=AD
D
AE
Step 2: Draw a vertical line through .
λ
D
Step 3: Draw a circle with center so that the central angle .
σ
S
∠ESB=2δ
Step 4: Let the intersections of and , if they exist, be and . Join , , and .
σ
λ
C
1
C
2
AC
1
AC
2
BC
1
BC
2
Then either of the two triangles or is a solution.
ABC
1
ABC
2
Step 5: Draw the triangle E (dashed line).
C
1
C
2
Verification
Let AB=α and =β.
∠C
1
∠ABC
1
Consider the triangle (the verification for the triangle is similar). Because is the right bisector of , =α, so the angle B=α-β. But this angle subtends the chord and is half of the central angle over , which is . So .
ABC
1
ABC
2
λ
AE
∠DEC
1
∠EC
1
EB
EB
2δ
δ=α-β