19. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Special Point of Intersection

c

1.4

D

0.75

δ

0.4

steps

1

2

3

4

This Demonstration constructs a triangle

ABC

given the length

c

of its base

AB

, the difference

δ

of the base angles and the point

D

, the intersection of

AB

and

CS

, where

S

is the circumcenter.

Construction

Step 1: Draw a segment

AB

of length

c

and a point

D

on

AB

between

A

and

B

. Let

E

be the midpoint of

AB

.

Step 2: Let

S

be the intersection of the perpendicular bisector of

AB

and the ray

ρ

through

D

that forms the angle

δ+π/2

with

AB

.

Step 3: Draw a circle

σ

with center

S

and radius

SA=SB

. Let

C

be the intersection of

ρ

and

σ

.

Step 4: Triangle

ABC

is a solution of the problem.

Verification

The circle

σ

is the circumcircle of triangle

ABC

, so

∠EDS=δ=α-β

.