 32. Construct a Triangle ABC Given the Length of AB, the Ratio of the Other Two Sides and a Line through C

Requires a Wolfram Notebook System

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

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration shows how to construct a triangle given the length of the side , the ratio of the other two sides and a line containing .

[more]

Construction

1. Draw the line and a line . On choose points and , such that .

2. Construct a triangle such that . Let be the point where the angle bisector at meets . Let be the point where the angle bisector of the outer angle at meets .

3. Let be the midpoint of . Draw the circle with center and radius .

4. The point is an intersection of and .

Verification

The triangles and have the same Apollonius circle.

[less]

Contributed by: Gerd Baron, Izidor Hafner, Marko Razpet and Nada Razpet (July 2018)
Open content licensed under CC BY-NC-SA

Details

The bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle. To divide the segment in the ratio , construct a triangle with legs and so that . Choose and with so that and . The foot of angle bisector at divides in the ratio . The locus of such points is the Apollonius circle of the triangle and is independent of . The radius of the Apollonius circle of is if . The radius depends only on and .

Reference

 E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins Publishers, 1989, pp. 21–22.

Snapshots   Permanent Citation

Gerd Baron, Izidor Hafner, Marko Razpet and Nada Razpet

 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. Send