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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
.
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