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prashant singh
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1. Prove that nine point centre is the mid point of orthocentre and circumcentre.
Mark44 said:In case anyone else wasn't familiar with the term nine-point center, here's a wiki article: https://en.wikipedia.org/wiki/Nine-point_center
@prashant singh, have you looked at this article?
haruspex said:I know the requirement is awkward for proofs, but on these homework forums you are supposed to show some attempt. Do you have any thoughts at all on the matter?
You could also try following some of the referenced links from the Wikipedia article.
I don't have any attempts , I don't know how to start but I know euler line and I have tried to link them but I didn't get the answer. Do you have hints.haruspex said:I know the requirement is awkward for proofs, but on these homework forums you are supposed to show some attempt. Do you have any thoughts at all on the matter?
You could also try following some of the referenced links from the Wikipedia article.
I didn't post the article because I thought it had a proof. I posted it because I thought it might help you do your proof. Part of that article gives the coordinates of the nine-point center. If you can get the coordinates of the other points you're concerned with, you're pretty much done.prashant singh said:Yes I read the whole article but there was no proof.
That's not how it works here. From the rules (https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/):prashant singh said:If you know the proof then tell me directly
You have been given some hints -- it's up to you to show some effort. So far, I haven't seen any effort on your part.Giving Full Answers:
On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given.
The nine point centre is a point of interest in a triangle that is constructed by taking the midpoints of the three sides, the feet of the altitudes, and the midpoints of the line segments connecting the orthocentre to each vertex. It is important in geometry because it has many properties and relationships with other points in a triangle, such as the circumcentre and the centroid.
The nine point centre and the Euler line are both points of interest in a triangle. The Euler line connects the orthocentre, the circumcentre, and the centroid, while the nine point centre is constructed by taking specific points on the Euler line. Additionally, the nine point centre is also the midpoint of the line segment connecting the orthocentre and the circumcentre, which is a property of the Euler line.
Yes, the nine point centre can be located outside of the triangle. In fact, it is only located inside the triangle if the triangle is acute. If the triangle is right, the nine point centre lies on the midpoint of the hypotenuse. If the triangle is obtuse, the nine point centre lies on the extension of the line segment connecting the orthocentre and the vertex opposite the obtuse angle.
The nine point centre has practical applications in the field of engineering, specifically in structural analysis. It can be used to determine the location of the centre of gravity of a triangle, which is important in designing stable structures. It can also be used to determine the location of the incenter of a triangle, which is useful in designing inscribed shapes.
No, the concept of the nine point centre can also be applied to other polygons, such as quadrilaterals and regular polygons. However, the properties and relationships with other points may differ in these cases. The nine point centre is most commonly associated with triangles due to its many interesting properties and its relationship with the Euler line.