The altitudes of a triangle are concurrent.

Let

,

and

be the vertices of the triangle

. If

is the slope of

, then using two point formula to find slope of line

Since altitude

is perpendicular to the side

, so its slope

is given as using condition of perpendicular slope is

Equation of altitude

passing through

with slope

is

For the equation of altitude

, we just replace

by

,

by

and

by

in (iii) (i.e.

), so

For the equation of altitude

, we just replace

by

,

by

and

by

in (iv) (i.e.

), so

To see whether the altitudes (iii), (iv) and (v) are concurrent, consider the determinant.

This shows that the altitudes of the triangle are concurrent.

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