Slope of AB = (3 - 0)/(4 - 7) = -1 Slope of BC = (4 - 0)/(10 - 7) = 4/3 Slope of CA = (4 - 3)/(10 - 4) = 1/6
Let AD be perpendicular to BC from A and BE be perpendicular to CA from B. Orthocentre is the point of intersection of AD and BE. Equation of AD: passing through (4, 3) and slope = -1/(slope of BC) = -3/4 y - 3 = (-3/4)(x - 4) 3x + 4y = 24
Equation of BE: passing through (7, 0) and slope = -1/(slope of CA) = -6 y - 0 = -6(x - 7) 6x + y = 42
Solve these equations for AD and BE and get the coordinates of orthocentre (48/7, 6/7).
Let the line L1 be the perpendicular bisector of AB and the line L2 be the perpendicular bisector of CA. Circumcentre is the point of intersection of L1 and L2.
Equation of L1: passing through mid-point (11/2, 3/2) of AB and slope = -1/(slope of AB) = 1 y - 3/2 = 1(x - 11/2) x - y = 4
Equation of L2: passing through mid-point (7, 7/2) of CA and slope = -1/(slope of CA) = -6 y - 7/2 = -6(x - 7) 12x + 2y = 91
orthocenter is the intersection of the 3 altitudes of a circle.
for example, the coordinates of the vertices of the triangle are A(a, b), B(c, d) and C(e, f)
first, get the equation of the line passing through C and is perpendicular to AB. (altitude to AB)
then, get the equation of the line passing through A and is perpendicular to BC. (altitude to BC)
then, find the intersection of these lines. (the altitude to AC will also pass through this point since the altitude of a triangle are concurrent to each other - meaning 3 or more lines intersecting at a point...)
the intersection is the coordinates of the orthocenter :D
centroid is the intersection of the medians of a triangle (median - line connecting a vertex of a triangle and the midpoint of the side opposite to it.) the coordinates of the midpoint of a triangle with vertices at (a, b), (c, d), (e, f) are ( (a+c+e)/3, (b+d+f)/3 )
circumcenter - the intersection of the 3 perpendicular bisectors of a triangle (perpendicular bisector - a line perpendicular to a side of a triangle passing through its midpoint) this point is equidistant to the vertices of the triangle.
btw, there is another "center" thing related to triangles...
incenter - intersection of the angle bisectors of a triangle (angle bisector - lines bisecting the angles of a triangle) the incenter is equidistant from the sides of the triangle :D
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