Show that the altitudes of a triangle are concurrent

Explanation

Let `ΔABC` be a triangle.

Also, let, `AD` and `BE` are two altitudes of `ΔABC` intersecting one another at `O`.

Now, to prove that the altitudes of `ΔABC` are concurrent. it is sufficient to prove that `CO` is perpendicular to `AB`.



Let, the position vectors of `A,B` and `C`, taking `O` as the origin, be given by `vec a, vec b` and `vec c` respectively.

Then, `vec(OA)=veca`, `vec(OB)=vecb` and `vec(OC)=vecc`

Now, since, `AD_|_BC`,

`=> vec(OA)_|_vec(BC)`

...