WOLFRAM|DEMONSTRATIONS PROJECT

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

​
R
0.8
δ
0.6
s
2.7
step 1
step 2
step 3
step 4
verification
This Demonstration constructs a triangle
ABC
given the circumradius
R
, the difference of angles
δ
at the base
AB
and the sum
s
of the lengths of the sides
BC
and
CA
.
Construction
Draw a (blue) circle
σ
1
with center
S
, radius
R
and a diameter
DE
. Let
C
be a point on
σ
1
such that
∠CED=δ/2
.
Step 1: Draw a (red) circle
σ
2
with diameter
CE
and center
M
, the midpoint of
CE
. Let
F
be on
σ
2
such that
CF=s/2
.
Step 2: Let
CF
intersect
σ
1
at
B
.
Step 3: Let
A
be symmetric to
B
across
DE
.
Step 4: Draw the triangle
ABC
.
Verification
Let
BC=a
and
CA=b
.
Let
G
be symmetric to
A
​
across
CE
. Let
P
be the intersection of
AB
​
and
CE
.
The angle between the angle bisector from
C
and the altitude from
C
is
δ/2
; also,
∠SCM=δ/2
.
So
CM
is the bisector of the angle at
C
. Since
CA=CG=b
,
CF=s/2
and
FG=FB
,
CB=a
, so
s=a+b
.