Concurrence of the Median, a Chord and a Diameter of the Incircle
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In the triangle 
, let the incircle have center 
and let its points of tangency to the sides 
, 
and 
be points 
, 
and 
, respectively. Let 
be the midpoint of 
. Then the lines 
, 
and 
are concurrent. Drag the locators 
, 
and 
to see that this holds for any triangle.
Contributed by: Sumith Nalabolu and Claire Wang (March 2020)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The incircle of a triangle is the circle contained in the triangle tangent to each of the three sides. Lines are concurrent if they all pass through a common point.
See [1] for the formulas AreaOfTriangle, Circumcircle and IntersectionofLines.
See [2] for the formulas Incircle and TangencyPoint.
This was a project for Advanced Topics in Mathematics II, 2019–2020, Torrey Pines High School, San Diego, CA.
References
[1] J. Warendorff. "The Second Lemoine Circle" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TheSecondLemoineCircle.
[2] J. Warendorff. "Collinearity of a Triangle's Circumcenter, Incenter, and the Contact Triangle's Orthocenter" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/CollinearityOfATrianglesCircumcenterIncenterAndTheContactTri.
Permanent Citation
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