Orthocenter

Elizabeth Nelli

We want to find the angles of DEF in terms of the angles of ABC.

First look at triangle AOC and triangle EOD. AO=EO because both are radii of the circle.
Similarly, CO=DO.
<AOC and <EOD are vertical angles and therefore congruent
Therefore, by SAS, triangle AOC is congruent to triangle EOD.
So we know, by corresponding parts, that AC=ED

The same can be said for triangles BOC and DOF.
BO=FO because both are radii of the circle
Similarly, CO=DO.
<DOF and <BOC are vertical angles and therefore congruent
Therefore, by SAS triangle BOC is congruent to triangle DOF.
So we know by corresponding parts that BC=DF

Now all we have left is to prove that AB=EF

Again, look at triangles AOB and EOF
EO=AO because both are radii of the circle
Similarly, FO=BO
<EOF and < AOB are vertical angles and therefore congruent
Therefore, by SAS triangle AOB is congruent to triangle EOF
So we know by corresponding parts that AB=EF

Therefore, we know that triangle ABC is congruent to triangle DEF and the corresponding angles of congruent triangles are congruent.

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