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

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

This Demonstration shows a 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 of length , and let be on the perpendicular to at such that .

AB

c

C'

AB

A

AG=b

Step 1: Construct the inscribed square of the triangle.

AD'DD''

Step 2: Construct a point at distance from and at distance from . Let the point be the intersection of and the circle with center and radius .

E

e=bc/(a+b)

D'

s

A

C

BE

A

b

Then satisfies the conditions.

ABC

Verification

Let the point be the fourth point of the rhombus . In any triangle , the inscribed rhombus has side length , which is independent of .

F

AD'EF

ABC

AD'EF

e=bc/(a+b)

∠BAC

By construction, and .

AB=c

AC=b

Since and is a diagonal of the rhombus , is the angle bisector of at of length .

AE=s

AE

AD'EF

AE

ABC

A

s