WOLFRAM|DEMONSTRATIONS PROJECT

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

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