# 4. Construct a Triangle Given Its Circumradius, Inradius and a Vertex Angle

4. Construct a Triangle Given Its Circumradius, Inradius and a Vertex Angle

This Demonstration constructs a triangle given its circumradius , inradius and the angle at .

ABC

R

r

γ

C

Construction

Draw a circle with center of radius and a chord of length . Let be the midpoint of and let be an endpoint of the diameter perpendicular to .

σ

1

S

R

AB

2Rsinγ

M

AB

D

AB

Step 1: Draw a line parallel to at distance from intersecting at .

p

AB

r

AB

SD

Z

Step 2: Draw a second circle with center through and . Let be one of the points of intersection of and .

σ

2

D

A

B

T

p

σ

2

Step 3: Let be the intersection of and .

C

σ

1

DT

Step 4: Draw the triangle and its incircle.

Verification

Theorem: Let . Then .

AB=c

c=2Rsinγ

Proof: In the right-angled triangle , the hypotenuse length is and the leg , so ).

Since the arc equals the arc , bisects . The distance of to is , so is the incenter of .

AMS

R

AM=c/2

sinγ=c/(2R

Since the arc

AD

DB

DC

∠ACB

T

AB

r

T

ABC