11453

2. Construct a Triangle Given the Circumradius, the Difference of the Base Angles, with the Circumcenter on the Incircle

This Demonstration shows a construction of a triangle given its circumradius , the difference of the base angles and that the circumcenter is on the incircle.
Let be the inradius of . The Euler formula gives the distance between the circumcenter and the incenter. Since the circumcenter is on the incircle, , which has the positive solution .
Construction
Draw a circle of radius with center and draw a diameter . Draw a chord at an angle from .
Step 1: Draw a circle with center and radius . Of the two points of intersection of and the segment , let the point be the one closest to .
Step 2: Draw a ray from at an angle from . Let be the perpendicular projection of on . Measure out a point on at distance from .
Step 3: The points and are the intersections of and the line through is perpendicular to .
Verification
Let , and .
Theorem: Let be any triangle. Let be the foot of the altitude from to , and let be the center of the circumscribed circle. Then the angle at between the altitude and equals . The angle between and the angle bisector at is . (See The Plemelj Construction of a Triangle 4.)
Proof of the last part: Let be on the angle bisector at , then .
By construction and the theorem, and is the circumscribed circle of triangle with center and radius .
By construction, is on the angle bisector at and the distance of to is . So the circle with center and radius is the incircle of , which by construction contains .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Reference
[1] D. S. Modic, Triangles, Constructions, Algebraic Solutions (in Slovenian), Ljubljana: Math Publishers, 2009 p. 83.
    • 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 »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+