WOLFRAM|DEMONSTRATIONS PROJECT

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

R

0.8

r

0.3

γ

0.79

step 1

step 2

step 3

step 4

This Demonstration constructs a triangle

ABC

given its circumradius

R

, inradius

r

and the angle

γ

at

C

.

Construction

Draw a circle

σ

1

with center

S

of radius

R

and a chord

AB

of length

2Rsinγ

. Let

M

be the midpoint of

AB

and let

D

be an endpoint of the diameter perpendicular to

AB

.

Step 1: Draw a line

p

parallel to

AB

at distance

r

from

AB

intersecting

SD

at

Z

.

Step 2: Draw a second circle

σ

2

with center

D

through

A

and

B

. Let

T

be one of the points of intersection of

p

and

σ

2

.

Step 3: Let

C

be the intersection of

σ

1

and

DT

.

Step 4: Draw the triangle and its incircle.

Verification

Theorem: Let

AB=c

. Then

c=2Rsinγ

.

Proof: In the right-angled triangle

AMS

, the hypotenuse length is

R

and the leg

AM=c/2

, so

sinγ=c/(2R

).Since the arc

AD

equals the arc

DB

,

DC

bisects

∠ACB

. The distance of

T

to

AB

is

r

, so

T

is the incenter of

ABC

.