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Thread: How can I construct an ellipse given one focus and two points on the ellipse?

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    How can I construct an ellipse given one focus and two points on the ellipse?

    I created the attached sketch to answer your question. Here is a snapshot from the sketch that I'll refer to in my explanation below. (Click on image to enlarge.)

    Click image for larger version

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    There is a custom ellipse tool that you can access by going to Help | Sample Sketches & Tools. It requires two foci and one point on the ellipse, so it took a little thinking and experimentation until I figured out how to construct the second focus from the three given points (one focus and two points on the ellipse). I used the property that the sum of the distances from any point on the ellipse to both foci is constant.

    The circle centered on focus 1 is used to find the distance along the segment from focus 1 to point B equal to the distance from focus 1 to point A. The segment that remains is used as the radius of the circle centered on point A (by using Construct | Circle by Center+Radius).

    This circle is used to find the point on the segment from point A to point B that represents the total distance from focus 1 to point B, along the path that goes from focus 1 to point A and then onward toward point B. The second focus is the midpoint of the segment that remains from the edge of that circle to point B. Then I used the custom ellipse tool using the two foci and either point on the ellipse.

    In Sketchpad, this diagram is dynamic. You can move focus 1 and both points (although point A must be closer to focus 1 than point B). Hope that helps.
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