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

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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.

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Construction

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.

Verification

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

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

Details

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

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