12. Construct a Triangle Given the Length of Its Base, the Perimeter and the Difference of the Base Angles

c

s

δ

step 1

step 2

verification

This Demonstration shows a construction of a triangle

ABC

given the length

c

of its base

AB

, the sum

s

of the lengths of the other two sides

BC

and

CA

, and the difference

δ

of the base angles.

Construction

Let

∠CAB=α

,

∠ABC=β

and

∠BCA=γ

.

Draw a line segment

DB

of length

s

.

Step 1: Draw a circle with center

S

with central angle

π+(α-β)

above the chord

BD

.

Step 2: Measure out the point

A

on the circle at distance

c

from the point

B

on the other side of

DB

from

S

. Let the point

C

be the intersection of

DB

and the right bisector of

AD

.

Verification

Since the central angle

DSB

equals

π+δ

by construction,

∠DAB=π/2+δ/2

. On the other hand,

∠DAB=α+γ/2=α+(π-α-β)/2=π/2+(α-β)/2

, so

δ=α-β

.