9. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Sum of the Other Two Sides

s

δ

c

step 1

step 2

verification

This Demonstration shows a construction of a triangle

ABC

given the length of the base

AB=c

, the sum

s

of the other two sides

BC

and

CA

, and the difference

δ

of the base angles.

Construction

Draw a line segment

BD

of length

s

.

Step 1: Draw a circle with center

S

so that the central angle

∠DSB=π-δ

. Measure out the point

A

on

σ

at distance

c

from

B

.

Step 2: Let the point

C

be the intersection of the perpendicular bisector of

AD

and

BD

.

Verification

Let

∠CAB=α

,

∠ABC=β

and

∠BCA=γ

.

Since

∠DSB=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

. Since

DAC

is isosceles,

AC+CB=DB

. By construction,

AB=c

.