# 33. Construct a Triangle Given Its Base, Altitude to the Base and Product of the Other Two Sides

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This Demonstration constructs a triangle given the length of the base , the length of the altitude from to and the product of the other two sides. Since the radius of the circumcircle is a rational function of and , we can construct it using similar triangles.

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Construction

2. Now construct the center of the circumcircle. There are two possibilities, one on either side of a horizontal line.

3. The point is the intersection of the dashed line through parallel to the horizontal line with either of the two circles. Since , , .

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

## Details

A Cassini oval (or Cassini ellipse) is a quartic curve such that if is on the curve, the product of its distances from two fixed points and at a distance apart is a constant . Thus the original problem is equivalent to finding the intersections of the oval with a line parallel to the axis at distance .

Keeping and fixed and changing , we get another construction of points on the Cassini oval.

## Permanent Citation

Izidor Hafner and Marko Razpet

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