Pythagoras' Theorem.

The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

Euclid's proof:

Another proof:

The medians of a triangle are concurrent at the centroid.

The altitudes of a triangle are concurrent at the orthocentre.

The internal bisectors of the angles of a triangle are concurrent at the centre of the incircle.

The perpendicular bisectors of the sides of a triangle are concurrent at the centre of the circumcircle.

Pappus' Theorem:

Pascal's Theorem:

The Simson Line

The Nine-Point Circle

The Euler Line

Morley's Theorem

The "outer" internal angle trisectors of a triangle intersect to form an equilateral triangle.

Napoleon's Theorem