WOLFRAM|DEMONSTRATIONS PROJECT

11b. Construct a Triangle Given the Lengths of Two Sides and the Bisector of Their Common Angle

​
b
c
s
δ
step 1
step 2
step 3
This Demonstration shows an alternative construction of a triangle
ABC
given the lengths of the sides
b
and
c
and the length
s
of the angle bisector of
∠BAC
.
Construction
Draw the line segment
DB
1
with interior point
A
1
so that

DA
1
=b
and

A
1
B
2
=c
.
Step 1: Draw parallel lines through
A
1
and
D
at an arbitrary angle
δ
. On the line through
A
1
, measure out
A
3
so that

A
1
A
3
=s
. Let
C
be the intersection of
B
1
A
3
and the parallel line through
D
.
Step 2: Construct an isosceles triangle
DCA
with base
DC
and a leg of length
b
.
Step 3: Let
A
2
be a point such that
A
A
2

A
1
A
3
and
A
A
2
=
A
1
A
3

. That is,
A
A
1
A
3
A
2
is a parallelogram. Let point
B
​
be the intersection of the lines
DA
and
CA
2
.
Verification
In the triangle
DBC
,
AA
2
divides
BC
in the ratio
b:c
. So
AA
2
is the bisector of the angle at
A
.