30. Construct a Triangle ABC Given the Length of AB, the Angle at C and the Difference of the Other Two Sides

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This Demonstration shows how to construct a triangle given the length of side , the opposite angle at and the difference of the lengths of the other two sides .



1. Draw the segment of length .

2. Draw the circle with center so that .

3. Draw the circle with center and radius . Let the circles and meet at .

4. Let the point be the intersection of the ray and the perpendicular bisector of .

5. Then is the required triangle.


since it is an arc corresponding to the central angle . Since is an isosceles triangle with apex , and .

Thus .


Contributed by: Izidor Hafner (April 2018)
Open content licensed under CC BY-NC-SA



The problem was posed in [1, section IX, problem 2, solution pp. 291–292] and in [2, problem 2, p. 143].


[1] M. Bland, Geometrical Problems Deducible from the First Six Books of Euclid, Arranged and Solved, Cambridge: J. Smith, 1819.

[2] M. Bland and A. Wiegand, Geometrische Aufgaben für Hohëre Lehranstalten, Braunschweig: Schewetschke und Sohn, 1865. babel.hathitrust.org/cgi/pt?id=njp.32101076800687;view=1up;seq=143.

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