# 6. Construct a Triangle Given Its Circumradius, Inradius and the Length of Its Base

6. Construct a Triangle Given Its Circumradius, Inradius and the Length of Its Base

This Demonstration constructs a triangle given the circumradius , inradius and the length of the base .

ABC

R

r

AB=c

Construction

Draw a circle of radius and a chord of length .

σ

1

R

AB

c

Step 1: Let be the midpoint of and let the right bisector of meet at the point .

D

AB

AB

σ

1

E

Step 2: Draw a second circle with center through and . Draw a (green) line perpendicular to at so that . Clearly is parallel to .

σ

2

E

A

B

p

ED

F

DF=r

p

AB

Step 3: Let be one of the points of intersection of and . The point is the intersection of and the ray .

G

p

σ

2

C

σ

1

EG

Verification

Let , and .

∠CAB=α

∠ABC=β

∠BCA=γ

Since arc equals arc , bisects and . Also . The triangle is isosceles, so . Then . So is the incenter of with distance from .

AE

EB

EC

∠ACB

∠BAE=γ/2

∠AEG=∠ABC=β

AEG

∠EAG=π/2-β/2

∠GAB=∠GAE-∠BAE=π/2-β/2-γ/2=α/2

G

ABC

r

AB