# 18. Construct a Triangle Given the Length of Its Base, the Difference of Angles at Its Base and the Point of Intersection of an Angle Bisector with Its Base

18. Construct a Triangle Given the Length of Its Base, the Difference of Angles at Its Base and the Point of Intersection of an Angle Bisector with Its Base

This Demonstration shows a construction of a triangle given the length of its base , the difference of the base angles and the point of intersection of the angle bisector with the base .

ABC

c

AB

δ

D

b

C

AB

Construction

Step 1: Draw a segment of length and a point on that segment.

AB

c

D

Step 2: Construct the ray from that forms the angle with and the ray from that forms the angle with . Let be the intersection of and .

σ

A

δ/2

AB

τ

D

δ

AB

E

σ

τ

Step 3: Draw a ray from at an angle from . The point is the intersection of with an extension of the segment .

ρ

D

π/2+δ/2

AB

C

ρ

BE

Step 4: Triangle is a solution of the problem.

ABC

Verification

The exterior angle of the triangle at is . So , .

DBE

E

α

α=β+δ

δ=α-β