Points of Concurrency on a Triangle: Quick Guide
Circumcenter = the point of concurrency of the perpendicular bisectors of a triangle.
To find the circumcenter:
● Find the coordinates of the midpoint of one of the triangle’s sides
● Find the slope of that same side of the triangle
● Find the negative reciprocal of that slope
● Plug the negative reciprocal slope & the coordinates of the midpoint into point-slope
formula — y-y1 = m(x-x1) — to find the equation of that perpendicular bisector in
y=mx+b form
● Repeat to find the equation of one other perpendicular bisector of the triangle
● Set the two y=mx+b equations equal to each other and solve for x (this will be the
x-value of the circumcenter)
● Plug that x-value back into one of the y=mx+b equations to find y (this will be the y-value
of the circumcenter)
Orthocenter = the point of concurrency of the lines containing the altitudes of a triangle.
*Note the orthocenter isn’t always inside the triangle.
To find the orthocenter:
● Find the slope of one of the triangle’s sides
● Find the negative reciprocal of that slope
● Plug the negative reciprocal slope & the coordinates of the vertex on the opposite side
into point-slope formula to find the y=mx+b equation of that altitude
● Repeat to find the equation of one other altitude
● Set the two y=mx+b equations equal to each other and solve for x (x-value of
orthocenter)
● Plug that x-value back into one of the y=mx+b equations to get the y-value of the
orthocenter
Centroid = the point of concurrency of the medians of a triangle.
To find the centroid:
● Find the coordinates of the midpoint on one of the triangle’s sides
● Plug the coordinates of the midpoint & the coordinates of the vertex on the opposite side
into point-slope formula to find the y=mx+b equation of that median
● Repeat to find the equation of one other median
● Set the two y=mx+b equations equal to each other and solve for x (x-value of centroid)
● Plug that x-value back into one of the y=mx+b equations to get the y-value of the
centroid