# 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