"A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such expert experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime." George Polya, 1944
 

HINTS:

CIRCLE is the set of all points in a plane that are at the same distance from a fixed point called the center.

Tangent of a circle is a line that touches the circle at one and only one point no matter how far produced.

Proposition. If a line is tangent to a circle, it is perpendicular to a radius at the point of tangency.

INCENTER of a triangle

Proposition: The bisectors AD, BF and CE of the angles of a triangle ABC meet in a point I, which is equidistant from the sides of the triangle.

The incircle is the inscribed circle of a triangle. The center of the incircle is called the incenter, and the radius of the circle is called the inradius.

PERPENDICULAR LINES AND ANGLES

Proposition: Two angles are congruent or supplementary if their sides are respectively perpendicular to each other.
 

AREA OF A TRIANGLE:

Proposition: The area of a triangle equals one-half the product of the length of a side and the length of the altitude to that side.

Side Angle Side Formula: The SAS formula = ½ (side1 × side2) × sine(included angle).