# The Plemelj Construction of a Triangle: 5

The Plemelj Construction of a Triangle: 5

This Demonstration constructs a triangle given the length of its base, the length of the altitude from to and the difference between the angles at and .

ABC

AB=c

h

C

C

AB

α-β

A

B

Construction

Step 1: Draw a line segment of length and let its midpoint be . Draw a line segment of length perpendicular to . Through , draw a straight line parallel to .

AB

c

M

AA'

h

C

AB

A'

ρ

AB

Step 2: Draw a circle so that the segment is viewed at angle as a chord of the circle, which implies the central angle .

σ

AB

δ

∠AOB=2δ

Step 3: Let be the intersection of the ray and the circle .

D

MA'

σ

Step 4: The point is the intersection of and .

C

DB

ρ

Step 5: The triangle meets the stated conditions.

ABC

Verification

This is the most elegant construction. It is based on the fact that the exterior angle of the triangle at is and the interior angle at is , so the angle at is .

ADB

A

α

B

β

D

α-β