I have a triangle ABC. The slope of AB is -1/ab, the slope of AC is -1/ac, and the slope of BC is -1/ac. My question is, I have 3 lines: Altitude from A to BC, altitude from B to AC, and altitude from C to AB. I know their slope because it is just -1/slope. But I want to know the coordinates of where they intersect. Could someone help me? thanks
The details get a bit messy, but you can place A=(0,0) B=(h,k) C=(m,n)
Then the altitude from B to AC is
y-k = -h/k (x-h)
The altitude from C to AB is
y-n = -m/n (x-m)
solve for x and y.
Trilinear coordinates are much cleaner, but less well known.
thanks a bunch
To find the coordinates of the intersection points of the three altitudes of a triangle, you need to determine the equations of the lines that the altitudes lie on. Once you have the equations, you can solve them simultaneously to find the intersection points.
To start, let's assume you have a triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3).
1. Altitude from A to BC:
The equation of a line passing through A with slope -1/(slope of BC) can be written as:
y - y1 = -1/(slope of BC) * (x - x1)
2. Altitude from B to AC:
The equation of a line passing through B with slope -1/(slope of AC) can be written as:
y - y2 = -1/(slope of AC) * (x - x2)
3. Altitude from C to AB:
The equation of a line passing through C with slope -1/(slope of AB) can be written as:
y - y3 = -1/(slope of AB) * (x - x3)
Now, you have three linear equations, and you can solve them simultaneously to find the intersection points.
To find the intersection point between the altitude from A and BC and the altitude from B to AC, solve equations 1 and 2 together. Similarly, solve equations 1 and 3 to find the intersection point between the altitude from A to BC and the altitude from C to AB. Finally, solve equations 2 and 3 to find the intersection point between the altitude from B to AC and the altitude from C to AB.