History of incenter and Euler line It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and Euler line.")
 A: It's true $-$ Euler was the first to show that if the incenter lies on the Euler line that the triangle is isosceles.  Euler's 1763 paper, Solutio facilis problematum quorundam geometricorum difficillimorum, is nicely discussed in Ed Sandifer's How Euler Did It: The Euler Line and Sandifer briefly discusses Euler's handling of the case where the orthocenter, centroid, circumcenter, and incenter are collinear on the last page of his summary article.  Euler's demonstration isn't what most people would consider elegant: Euler associates the lengths of the triangle sides to a particular cubic equation and shows in the case the four points are collinear that the cubic equation has a double root.  But this particular result is just a minor side note in Euler's analysis related to determining the sides of a triangle given the four points.  
The actual paper, in the original Latin, is here at E325.  Google Translate has a good Latin translator in case your Latin is a little rusty. :)  Euler's presentation of the double root for the cubic when the four points are collinear is in section 26 of the paper ($a=b=\ldots$)
