A triangle PQR whose vertices are rational points which of the following points of the triangle are always rational pointsA.centroidB. incentreC.circumcentreD.orthocentre

Dear yoganshu,

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 

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Aman Bansal

Askiitian Expert

othocentre , centroid and circumcentre....but incentre need not "always" be a rational point.......best of luck