Independent+Project+(AD+&+SmJ)

 __ Introduction __

The triangle is a shape that has long been an object of fascination for mathematicians and other scientists alike. It’s a figure with properties so unique, that its applications probably number in the millions. From buses to buildings, we use these magical figures to improve our everyday lives. Inspired by the triangle, me and Seung-Mo decided to explore an area that we weren’t very familiar with- the centers of a triangle.

__ Profile: Leonhard Euler __

As part of our project, we looked at the Euler line, a line discovered by Leonhard Paul Euler, a Swiss mathematician and physicist who spent most of his life in Russia and Germany. Most would consider him to be the greatest mathematician of the 18th century, and he definitely makes the list of the most prolific mathematicians of all time. Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, optics, and astronomy. Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. If printed, his works would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics. Euler introduced and popularized several notational conventions through hProxy-Connection: keep-alive Cache-Control: max-age=0 numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter // i // to denote the imaginary unit. Euler also popularized the use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter, although it did not originate with him. Surprisingly, he did much of his work after he became blind in one eye in 1735 and totally blind in 1766.




 * Circumcenter **

The circumcenter of a triangle is the center of its circumcircle. For a polygon that can be circumscribed, its found at the point where the three perpendicular bisectors intersect (bisector meaning line going through the midpoint of a side, and thus the perpendicular bisector would be the bisector of a line segment hitting it at a right angle/90 degrees). The circumcenter of a triangle can be outside the triangle (in obtuse triangles). Circumcircle is the circle that touches/meets all 3 verticies of a triangle. And circumradius is the radius of the circumcenter. Where its found in... Acute triangles- inside of the triangle Right triangles- Midpoint of the hypotenuse Obtuse triangles- outside of the triangle ** Incenter ** The incenter of a triangle is the point that’s right at the center of a triangle’s incirlce. You can find this point by finding the point at which the angle bisectors of a triangle intersect (a ray that splits an angle into two equal halves.). The incenter is always on the inside of a triangle. The incircle is the circle that touches/meets the 3 sides of the triangle. The inradius is the radius of the incircle. Position: 1. incenter is always in the inside of the triangle 2. isosceles triangles’ incenter and circumcenter are on the bisector of the apex 3. Regular triangles’ incenter and circumcenter are the same, as the incircle and circumcircle are essentially two circles with the same centers but with dufferent radii

** The centroid of a triangle is the point at which the three medians of a triangle intersect. A median is a line segment from a vertex to the midpoint of the opposite side, and always divides the triangle into two equal halves. The centroid of a triangle is always inside the triangle. ** The orthocenter of a triangle is the point at which the three (possibly extended) altitudes (a line segment from a vertex to the point at which it strikes the opposite side at an angle of 90 degrees). For an acute triangle, it will always be on the interioir; for a right triangle, it will always be the right angle itself; and for obtuse triangles, it will always be outside the triangle. Where its found in... Acute triangles -inside Right triangles - right-angled vertex Obtuse triangles -outside
 * Centroid (also referred to as the ‘center of mass/gravity’)
 * Orthocenter

** Excenter **

Excenter is the center of excircle (all three excircles). Excircle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. The exradius is the radius of the excircle.There are always 3 excenters in a triangle. The excenter of a triangle is always in the outside of the triangle. To find the excenter of a triangle, we first extend all three sides, and find the angle bisectors of each of the newly formed angles. The distance between the point where the two bisectors intersect and the point on the triangle directly perpendicular to it becomes the exradius, which makes the point foun earlier the excenter. (Remember, this process is repeated with all three sides of a triangle)


 * Euler Line**

Euler line is the line determined from any triangle that is not equilateral; it passes through the orthocenter, circumcenter, and the centroid. (Named after Leonhard Euler). On an equilateral triangle, all 3 centers coincide or meet at the same spot. Other notable points that lie on the Euler line are the ‘__de Longchamps point__’, ‘__the Schiffler point__’, ‘__the Exeter point__’, ‘__the far-out point__’, and ‘__the nine-point center__’. media type="custom" key="2715845" media type="custom" key="2715863"