34. Construct a Triangle ABC Given the Length of AB and the Sum and Product of the Other Two Sides

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows how to construct a triangle given the length of the side , the sum of the lengths of the other two sides and their product . The numbers and satisfy the quadratic equation . It is solved graphically using Descartes's method for constructing roots of polynomials.



1. Draw a vertical line and a horizontal line .

2. Through , draw a vertical line .

3. Draw a circle with center and radius . Let and be the points of intersection of the circle with .

4. Draw points and on a horizontal line through such that and is the midpoint of .

5. Let the point be the intersection of the circle with center and radius and the circle with center and radius .

6. The triangle satisfies the given conditions.


According to Descartes's method, and are the solutions of the quadratic equation, but and , so and . By step 5, .


Contributed by: Izidor Hafner and Marko Razpet (September 2018)
Open content licensed under CC BY-NC-SA


Let . Thus or . The triangle inequality is , which is . Suppose ; the triangle inequality implies .


Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.