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.
THINGS TO TRY
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?
the Wolfram Demonstrations Project
Embed Interactive Demonstration
More details »
Download Demonstration as CDF »
Download Author Code »
More by Author
The Johnson Circles
The Conway Circle
Circles through the Orthocenter
Three Circles with Two Common Tangents
Intersecting Circles and a Right Angle
Inscribing Four Circles in a Triangle
The Brocard Points and Neuberg Circles of a Triangle
Twins of Arbelos and Circles of a Triangle
Problems on Circles IX: Circumcircles and Incircles of a Triangle
Browse all topics
The #1 tool for creating Demonstrations
and anything technical.
Explore anything with the first
computational knowledge engine.
The web's most extensive
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.
Join the initiative for modernizing
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
© 2013 Wolfram Demonstrations Project & Contributors |
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