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

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

This Demonstration shows an alternative construction of a triangle given the lengths of the sides and and the length of the angle bisector of .

ABC

b

c

s

∠BAC

Construction

Draw the line segment with interior point so that and .

DB

1

A

1

=b

DA

1

=c

A

1

B

2

Step 1: Draw parallel lines through and at an arbitrary angle . On the line through , measure out so that . Let be the intersection of and the parallel line through .

A

1

D

δ

A

1

A

3

=s

A

1

A

3

C

B

1

A

3

D

Step 2: Construct an isosceles triangle with base and a leg of length .

DCA

DC

b

Step 3: Let be a point such that and . That is, is a parallelogram. Let point be the intersection of the lines and .

A

2

A

A

2

A

1

A

3

A=

A

2

A

1

A

3

A

A

1

A

3

A

2

B

DA

CA

2

Verification

In the triangle , divides in the ratio . So is the bisector of the angle at .

DBC

AA

2

BC

b:c

AA

2

A