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28. Construct a Triangle ABC Given the Length of AB, the Sum of the Other Two Sides, and a Line Containing C

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This Demonstration shows how to construct a triangle given the length of the side opposite the vertex , the sum of the lengths of the other two sides and a line containing the vertex . Since lies on the ellipse with foci and , the semi-axes of the ellipse are and .

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Let be the circle with center at the midpoint of and of radius . Let the points and be on the perpendicular bisector of at heights and , respectively. Let and be the lines parallel to through and . Let . Let the line through perpendicular to meet at . Let the line intersect at . Then the point lies on the line perpendicular to through such that the ratio of the distances of and from is .

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

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This construction is suggested by the linear transformation , which maps the ellipse and the line into the circle and the line .

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