# 9. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Sum of the Other Two Sides

9. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Sum of the Other Two Sides

This Demonstration shows a construction of a triangle given the length of the base , the sum of the other two sides and , and the difference of the base angles.

ABC

AB=c

s

BC

CA

δ

Construction

Draw a line segment of length .

BD

s

Step 1: Draw a circle with center so that the central angle . Measure out the point on at distance from .

S

∠DSB=π-δ

A

σ

c

B

Step 2: Let the point be the intersection of the perpendicular bisector of and .

C

AD

BD

Verification

Let , and .

∠CAB=α

∠ABC=β

∠BCA=γ

Since by construction, . On the other hand, , so . Since is isosceles, . By construction, .

∠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

DAC

AC+CB=DB

AB=c