Which point(s) of concurrency in a triangle is (are)never outside the triangle?
Is the orthocenter the only concurrency point that stays inside a triangle?
Actually the orthocentre could be either inside or outside,
if acute angled triangle, ---> inside
if obtuse angled triangle --> outside
http://www.google.ca/imgres?imgurl=http://www.mathwords.com/o/o_assets/orthocenter_obtuse.jpg&imgrefurl=http://www.mathwords.com/o/orthocenter.htm&h=294&w=340&sz=14&tbnid=VX63Op8LOppG1M:&tbnh=103&tbnw=119&prev=/images%3Fq%3Dorthocenter&hl=en&usg=__kUd6Go8Yc4_u7ltGTxIIhqTafE0=&ei=8j7pS6_kIMP58AbPlq3sDg&sa=X&oi=image_result&resnum=8&ct=image&ved=0CD0Q9QEwBw
the incentre, which is the angle bisector intersection, and the centroid are the only ones that stay inside
No, the orthocenter is not the only concurrency point that stays inside a triangle. In fact, there are three points of concurrency in a triangle that are always inside the triangle. These points are as follows:
1. Centroid: The centroid is the point of intersection of the triangle's medians. A median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. To find the centroid, you can locate the midpoints of all three sides of the triangle, and then connect these midpoints to their corresponding opposite vertices. The intersection point of these lines is the centroid.
2. Circumcenter: The circumcenter is the point of intersection of the perpendicular bisectors of the triangle's sides. A perpendicular bisector is a line that cuts a line segment into two equal halves while also being perpendicular to that line segment. To find the circumcenter, you need to locate the midpoint of each side of the triangle and draw a line that is perpendicular to that side and passes through its midpoint. Repeat this process for all three sides, and the point where these three lines intersect is the circumcenter.
3. Incenter: The incenter is the point of intersection of the triangle's angle bisectors. An angle bisector is a line that divides an angle into two equal parts. To find the incenter, draw the angle bisector for each of the triangle's angles. The point where these three angle bisectors intersect is the incenter.
It is worth noting that while the orthocenter, in general, is not always inside a triangle, it can be inside the triangle depending on the type of triangle (e.g., acute-angled triangle). However, the centroid, circumcenter, and incenter are guaranteed to be inside the triangle regardless of its type.