Lucas Circles

Given a triangle, find three circles that are tangent to the circumcircle of the triangle at the three points and that are also tangent to each other.
Construct three squares with two vertices on a side and the remaining two vertices on each remaining side. The squares are readily found by drawing the altitudes and using similarity of the triangles.
The three pairs of square vertices form three triangles with the triangle vertices. The circumcircles of the triangles are the required circles.



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


Was Édouard Lucas (1842–1891) the inventor of this gimmick? The problem evokes Descartes's theorem or Soddy circles. Who would think of using these easy-to-construct squares, which apparently do not seem to be related to the problem?
    • 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+