13. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Difference of the Lengths of the Other Two Sides
13. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Difference of the Lengths of the Other Two Sides
This Demonstration shows a construction of a triangle given the length of the base , the difference of the lengths of the other two sides and , and the difference of angles at the base.
ABC
c
AB
d
BC
CA
δ
Construction
Step 1: Draw a line of length and a circle with center and central angle over the chord . Measure out the point on the circle at distance from .
BE
d
S
δ
BE
A
c
B
Step 2: Let the point be the intersection of and the right bisector of .
C
BE
AE
Then satisfies the given conditions.
ABC
Verification
Let and .
∠CAB=α
∠ABC=β
By construction, . By step 2, , so , the difference of the other sides.
AB=c
CA=CE
d=BE=BC-CE=BC-CA
Since the central angle by construction, . The line bisects the angle at . In any triangle, the , ; since is isosceles, . So and .
∠ESB=δ
∠EAB=δ/2
DC
C
∠DEC=∠DAC=α
∠EDB=α-β
ADE
∠EAB=(α-β)/2
δ/2=(α-β)/2
δ=α-β