11562

# 20b. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Line Containing the Third Vertex

This Demonstration constructs a triangle given the length of its base, the difference of the base angles and a line (given by a point and slope ) that contains . This generalizes the problem given to Plemelj when he was in high school, namely, the problem in which the line is parallel to . Our construction is a simple adaptation of The Plemelj Construction of a Triangle: 4.
Construction
Step 1: Draw a line segment of length and its midpoint . Draw a line .
Step 2: Draw a line segment from that is perpendicular to . It meets the line at .
Step 3: Choose any point on the ray .
Step 4: Let be on such that .
Step 5: Let be the circle with center and radius .
Step 6: Let be the intersection of and the ray . Join and .
Step 7: The point is the intersection of and the line through parallel to .
Step 8: The triangle meets the stated conditions.
Verification
Theorem: Let be any triangle, and let be the foot of the altitude from to . Let be the circumcenter of , and let the segment be perpendicular to with on the same side of as . Then .
Proof: The inscribed angle subtended by the chord is , so the central angle . Since is isosceles, . Since , . Since , [1, Proposition 29]. So .
Now in the construction, let be the intersection of and the line through parallel to . Then the isosceles triangles and are similar, and is parallel to , so . By construction, , so . Since is on the right bisector of , it is the circumcenter of . By the theorem, .

### DETAILS

This problem comes from [2], where some suggestions on how to solve it are also given. That it could be solved by Plemelj's construction was observed by Nada and Marko Razpet. For an algebraic solution, see 20. Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles and a Line Containing the Third Vertex.
For the history of Plemelj's problem, references and a photograph of Plemelj's first solution, see The Plemelj Construction of a Triangle: 1.
References
[1] Euclid, Elements, Vol. 1, 2nd ed. (T. Heath, ed.), New York: Dover Publications, 1956 pp. 311–312.
[2] E. Heis and T. J. Eschweiler, Lehrbuch der Geometrie zum Gebrauche an höheren Lehranstalten, Köln: Verlag der M. Dumont-Schauberg'schen Buchhandlung, 1855.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.