# What is the orthocenter of a triangle with corners at #(4 ,7 )#, #(8 ,2 )#, and (5 ,6 )#?

Orthocenter coordinates

Slope of line segment BC

Slope of

Equation of altitude passing through A and perpendicular to BC

Slope of line segment AC

Slope of altitude BE perpendicular to BC

Equation of altitude passing through B and perpendicular to AC

Solving Eqns (1), (2) we arrive at the coordinates of orthocenter O

Coordinates of orthocenter

Verification :

Slope of

Equation of Altitude CF

Orthocenter coordinates

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Orthocenter:

I worked out the semi-general case [here].(https://tutor.hix.ai)

Let's test it by applying it to this triangle and comparing the result to the other answer.

First we translate (5 ,6) to the origin, giving the two other translated vertices:

We apply the formula in the translated space:

Now we translate back for our result:

That matches the other answer!

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