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

2



A

1

A

3

and

A

A

2

=

A

1

A

3



. That is,

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

.