12. Construct a Triangle Given the Length of Its Base, the Perimeter and the Difference of the Base Angles
12. Construct a Triangle Given the Length of Its Base, the Perimeter and the Difference of the Base Angles
This Demonstration shows a construction of a triangle given the length of its base , the sum of the lengths of the other two sides and , and the difference of the base angles.
ABC
c
AB
s
BC
CA
δ
Construction
Let , and .
∠CAB=α
∠ABC=β
∠BCA=γ
Draw a line segment of length .
DB
s
Step 1: Draw a circle with center with central angle above the chord .
S
π+(α-β)
BD
Step 2: Measure out the point on the circle at distance from the point on the other side of from . Let the point be the intersection of and the right bisector of .
A
c
B
DB
S
C
DB
AD
Verification
Since the central angle equals by construction, . On the other hand, , so .
DSB
π+δ
∠DAB=π/2+δ/2
∠DAB=α+γ/2=α+(π-α-β)/2=π/2+(α-β)/2
δ=α-β