Concurrence of the Median, a Chord and a Diameter of the Incircle
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.
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  for the formulas AreaOfTriangle, Circumcircle and IntersectionofLines.
See  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.
 J. Warendorff. "The Second Lemoine Circle" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TheSecondLemoineCircle.
 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.