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Prove that the altitudes of a triangl...

Prove that the altitudes of a triangle are concurrent.

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Let the vertices of a triangle be, `O(0,0)` `A(a,0)` and `B(b,c)` equation of altitude `BD` is `x=b`.
Slope of `OB` is `(c )/(b)`.
Slope of `AF` is `-(b)/(c )`.
Now, the equation of altitude `AF` is
`y-0=-(b)/(c )(x-a)`
Suppose, `BD` and `OE` intersect at `P`.
Coordinates of `P` are `[b,b(((a-b))/(c ))]`
Let `m_(1)` be the slope of `OP=(a-b)/(c )`
and `m_(2)` be the slope of `AB=(c )/(b-a)`
Now, `m_(1)m_(2)=((a-b)/(c ))((c )/(b-a))=-1`
We , get that the line through `O` and `P` is perpendicular to `AB`.
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