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

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.
1. On a horizontal line, draw points and so that and let be in the interval such that . Let be the midpoint of . On the vertical line through , draw the point so that . On the opposite side of the line , draw the points and so that and . Construct a point on the horizontal line so that is parallel to . Then . Draw a point on so that is parallel to . Then .
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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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.
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