29. Construct a Triangle ABC Given the Length of AB, the Angle at C and the Sum of the Other Sides

c

1

a + b

1.45

γ

1.3

zoom

2

steps

1

2

3

4

5

second solution

ellipse

verification

This Demonstration shows how to construct a triangle

ABC

given the length

c

of side

AB

, the opposite angle

γ

at

C

and the sum of the lengths of the other two sides

a+b

.

Construction

1. Draw the segment

AB

of length

c

.

2. Draw the circle

τ

with center

S

so that

∠ASB=γ

.

3. Draw the circle

σ

with center

A

and radius

a+b

. The circles

τ

and

σ

meet at

D

and

D'

.

4. Let the point

C

be the intersection of the segment

AD

and the perpendicular bisector of

BD

.

5. Then

ABC

is the required triangle.

Verification

∠ADB=γ/2

since it is an arc corresponding to the central angle

∠ASB=γ

.

Since

BDC

is an isosceles triangle with apex

C

,

∠ACB=γ

and

BC=CD

.

Thus

AC+CB=AC+CD=a+b

.

Details

The problem was posed in[1, section IX, problem 8, solution p. 295].

References

[1] M. Bland, Geometrical Problems Deducible from the First Six Books of Euclid, Arranged and Solved, Cambridge: J. Smith, 1819.

External Links

14. Construct a Triangle Given the Length of the Altitude to the Base, a Base Angle and the Sum of the Lengths of the Other Two Sides

9. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Sum of the Other Two Sides

24a. Construct a Triangle Given the Length of the Altitude to the Base, the Difference of Base Angles and the Sum of the Lengths of the Other Sides

Permanent Citation

Izidor Hafner

"29. Construct a Triangle ABC Given the Length of AB, the Angle at C and the Sum of the Other Sides" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/29ConstructATriangleABCGivenTheLengthOfABTheAngleAtCAndTheSu/
Published: April 3, 2018