Geometric Points of Concurrency

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Good Morning,  

As one of the premiere mathematicians on Centers of Triangles, your missions, should you choose to accept them, involve making your uncle happy, assisting an important Smithsonian archeological excavation in Cambodia, improving the quality of life for many citizens in rural Georgia, and, oh yes, preventing a cataclysmic collision between a destructive asteroid and our planet Earth. 

Your teacher who is also a covert member of the Mission Impossible Force (MIF) will select two team members to assist you on these critical missions.  You have forty-eight hours to receive your exact assignments and complete all four missions. As always, should any member of your team be caught or killed, the Secretary will disavow all knowledge of your actions.

This message may self-destruct in five seconds, so read the tasks carefully.