# 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