WOLFRAM|DEMONSTRATIONS PROJECT

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
.