SOLUTION: An equilateral triangle MPS has a point O inside. If MO=3, PO=4, and SO=5, determine the area of the equilateral triangle.

Question 1146324: An equilateral triangle MPS has a point O inside. If MO=3, PO=4, and SO=5, determine the area of the equilateral triangle.
Answer by greenestamps(13195) (Show Source): You can put this solution on YOUR website!

This is one of my favorite problems. When I first saw it many, many years ago, I spent several hours over the space of a couple of years trying different ways of solving it, finally coming up with an ugly solution that required a couple of pages of very ugly calculations. Then I discovered a path to the answer that takes only a few relatively easy steps.

Draw a picture of the equilateral triangle MPS with interior point O, using the given information. (In my picture, MS is horizontal with M at the left, with P above MS.)

Make a copy of triangle MOP (MO'P') and rotate it 60 degrees clockwise about point M so that MP' coincides with MS. We now have triangle MO'S with point O' below MS.

Angle PMS is 60 degrees, so angle OMO' is 60 degrees. And the lengths of MO and MO' are both 3, so triangle OMO' is equilateral, making angle MO'O 60 degrees.

Furthermore, triangle SOO' has side lengths 3, 4, and 5, making angle SO'O 90 degrees.

So angle MO'S is 150 degrees.

And now we can find the length of MS, the side of equilateral triangle MPS, using the law of cosines; and then we can find the area of triangle MPS using the formula for the area in terms of the length of a side.

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