19. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Special Point of Intersection
19. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Special Point of Intersection
This Demonstration constructs a triangle given the length of its base , the difference of the base angles and the point , the intersection of and , where is the circumcenter.
ABC
c
AB
δ
D
AB
CS
S
Construction
Step 1: Draw a segment of length and a point on between and . Let be the midpoint of .
AB
c
D
AB
A
B
E
AB
Step 2: Let be the intersection of the perpendicular bisector of and the ray through that forms the angle with .
S
AB
ρ
D
δ+π/2
AB
Step 3: Draw a circle with center and radius . Let be the intersection of and .
σ
S
SA=SB
C
ρ
σ
Step 4: Triangle is a solution of the problem.
ABC
Verification
The circle is the circumcircle of triangle , so .
σ
ABC
∠EDS=δ=α-β