# 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