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

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

This Demonstration constructs a triangle given the circumradius , the difference of angles at the base and the sum of the lengths of the sides and .

ABC

R

δ

AB

s

BC

CA

Construction

Draw a (blue) circle with center , radius and a diameter . Let be a point on such that .

σ

1

S

R

DE

C

σ

1

∠CED=δ/2

Step 1: Draw a (red) circle with diameter and center , the midpoint of . Let be on such that .

σ

2

CE

M

CE

F

σ

2

CF=s/2

Step 2: Let intersect at .

CF

σ

1

B

Step 3: Let be symmetric to across .

A

B

DE

Step 4: Draw the triangle .

ABC

Verification

Let and .

BC=a

CA=b

Let be symmetric to across . Let be the intersection of and .

G

A

CE

P

AB

CE

The angle between the angle bisector from and the altitude from is ; also, .

C

C

δ/2

∠SCM=δ/2

So is the bisector of the angle at . Since , and , , so .

CM

C

CA=CG=b

CF=s/2

FG=FB

CB=a

s=a+b