# What is the orthocenter of a triangle with vertices at #O(0,0 )#, #P(a,b)#, and Q(c,d)#?

Rather than asking a new question, I've generalized this old one. I've done this for a circumcenter question previously, and nothing went wrong, so I'm continuing the series.

As before, I tried to keep the algebra tractable by placing one vertex at the origin; an arbitrary triangle can be translated and the result can be translated back with ease.

The orthocenter is where a triangle's altitudes intersect. Its existence is predicated on the theorem that states that a triangle's altitudes intersect at a specific point, where we refer to the three altitudes as concurrent.

Let's demonstrate that the triangle OPQ's altitudes are concurrent.

The altitude from OP to Q has the following parametric equation:

From OQ to P, the altitude is similarly

Now let's examine where the altitudes from OP and PQ meet:

Adding,

Very nice to have the cross product in the denominator and the dot product in the numerator.

We have determined the orthocenter's coordinates and provided justification for calling the common intersection by that name.

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To find the orthocenter of a triangle with vertices at O(0,0), P(a,b), and Q(c,d), you can follow these steps:

- Calculate the slopes of the lines OP, OQ, and PQ using the coordinates of the given points.
- Determine the equations of the altitudes of the triangle passing through each vertex.
- Find the intersection point of these altitudes, which is the orthocenter of the triangle.

Let's denote the slopes as follows:

- Slope of OP: ( m_{OP} = \frac{b - 0}{a - 0} = \frac{b}{a} )
- Slope of OQ: ( m_{OQ} = \frac{d - 0}{c - 0} = \frac{d}{c} )
- Slope of PQ: ( m_{PQ} = \frac{d - b}{c - a} = \frac{d - b}{c - a} )

Now, the equations of the altitudes passing through O, P, and Q respectively are:

- Altitude through O: ( y = \frac{a}{b} x )
- Altitude through P: ( y - b = \frac{c - a}{d - b}(x - a) )
- Altitude through Q: ( y - d = \frac{a - c}{b - d}(x - c) )

Solve these equations simultaneously to find the intersection point, which gives the coordinates of the orthocenter of the triangle.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do I find the equation of a perpendicular bisector of a line segment with the endpoints #(-2, -4)# and #(6, 4)#?
- What is the centroid of a triangle with corners at #(6, 1 )#, #(2, 2 )#, and #(1 , 6 )#?
- What is the orthocenter of a triangle with corners at #(2 ,2 )#, #(5 ,1 )#, and (4 ,6 )#?
- A triangle has corners A, B, and C located at #(2 ,7 )#, #(3 ,5 )#, and #(9 , 6 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- What is the orthocenter of a triangle with corners at #(7 ,8 )#, #(3 ,4 )#, and (8 ,3 )#?

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