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

c

1.4

D

0.6

δ

0.8

steps

1

2

3

4

This Demonstration shows a construction of a triangle

ABC

given the length

c

of its base

AB

, the difference

δ

of the base angles and the point

D

of intersection of the angle bisector

b

C

with the base

AB

.

Construction

Step 1: Draw a segment

AB

of length

c

and a point

D

on that segment.

Step 2: Construct the ray

σ

from

A

that forms the angle

δ/2

with

AB

and the ray

τ

from

D

that forms the angle

δ

with

AB

. Let

E

be the intersection of

σ

and

τ

.

Step 3: Draw a ray

ρ

from

D

at an angle

π/2+δ/2

from

AB

. The point

C

is the intersection of

ρ

with an extension of the segment

BE

.

Step 4: Triangle

ABC

is a solution of the problem.

Verification

The exterior angle of the triangle

DBE

at

E

is

α

. So

α=β+δ

,

δ=α-β

.