16. Construct a Triangle Given Its Circumradius, Inradius and the Difference of Its Base Angles

This Demonstration shows a construction of a triangle given its circumradius , inradius and the difference of the base angles.
Draw a circle of radius with center and a diameter . (The circle will be the circumcircle of .)
Step 1: Draw a line segment at an angle from to meet at . From , draw a ray at angle from .
Let be on at the Euler distance from , where .
Step 2: Drop a perpendicular from to at and let be on such that .
Step 3: The points and are the intersections of and the perpendicular to at .
By construction, is the circumradius of .
Let , and .
Theorem: Let be the circumcenter of a triangle . If , the angle between the angle bisector at and is .
Proof: The angle subtended over from a point on to the right of is because . So the central angle is twice that: . Triangle is isosceles, so . It follows that
By this theorem and the construction of , is the angle bisector at . Therefore by Euler's triangle formula for , is the incenter of . Since is parallel to and , the distance from to is also and the incircle has center , radius .
By step 1, . The theorem states , so .
The distance is the geometric mean of and , that is of and .


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


[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.

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 © 2018 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+