WOLFRAM|DEMONSTRATIONS PROJECT

25. Construct a Triangle Given Its Base, the Difference of the Base Angles and the Length of One of Three Line Segments

1.85

0.4

1. altitude

2. angle bisector

3. circumcenterline segment

length

1.1

This Demonstration constructs a triangle

ABC

given the length

of its base

, the difference

between the angles

and

of the base and the length of one of the following:

i. the altitude from

to the base

ii. the bisector from

to the base

iii. the line segment through the circumcenter from

to the base

The last two cases are reduced to the first case, since the angle between the bisector and the altitude is

(α-β)/2

, and the angle between the altitude and the line through the center is

α-β

. So we have a Plemelj triangle, for which we use The Plemelj Construction of a Triangle: 13.

Construction

1. Draw line

of length

2. Draw a circle

of radius

c/2

with center

, the midpoint of

3. Draw a ray from

at an angle

to intersect

at the point

. Let

be a point such that the length of

is the length of the given altitude and

is perpendicular to

4. Draw a circle

with center

and radius

. Let the point

be on the circle such that

AP=AD

. The point

is the intersection of the perpendicular bisector of

and the line parallel to

through

Verification

The angles

∠DAC

and

∠CAP

are equal to

. The angle at

α=β+δ

The first case was posed to Josip Plemelj (1873–1967) by his mathematics teacher in secondary school in 1891 (in Ljubljana, then in the Austro-Hungarian Empire, now in Slovenia).