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