Prove: The altitudes of a triangle are concurrent.

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Given triangle ABC, we wish to demonstrate that altitudes BF, AE and CD are concurrent.

Note that

,

so

and

.

Similarly,

so

and

,

and

so

and

.

By substitution,

.

After cancellation,

,

and by the converse of Ceva's theorem, we have demonstrated that altitudes AE, BF and CD of triangle ABC are concurrent.

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