"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.

"Four phases trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook look is different when we have made some progress; it is again different when we have almost obtained the solution. In order to group conveniently the questions and suggestions of our list, we shall distinguish four phases of the work. First we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown-known is linked to the data in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it." George Polya, 1944.

Exercise your brain. Solve these problems about congruence of line segments, angles, triangles, circumferences, and similarity and lift up your geometry skills. Designed for high-school students, college geometry, SAT preparation and teachers with an interest in geometry problem-solving.
 

Table of Content

	Geometry Help: Key theorems or Facts you should know for the proposed problems.
	Proposed Problem 76: Area of a Circle. Square, Circle, Circular Sector.
	Proposed Problem 75: Three Intersecting Circles. Cyclic quadrilateral, Angles.
	Proposed Problem 74: Three Intersecting Circles. Cyclic quadrilateral, Angles.
	Proposed Problem 73: Three Intersecting Circles. Cyclic quadrilateral.
	Proposed Problem 72: Intersecting Circles. Cyclic quadrilateral, Chords, Parallel.
	Proposed Problem 71: Cyclic Quadrilateral.
	Proposed Problem 70: Squares inscribed in a triangle, Similarity.
	Proposed Problem 69: Square inscribed in a triangle, Similarity.
	Proposed Problem 68: Triangle, Incircle, Inradius, Tangent, Similarity.
	Proposed Problem 67: Triangle, Circumcircle, Angles, Cyclic Quadrilateral.
	Proposed Problem 66: Triangle, Excircle, Tangents, Geometric Mean.
	Proposed Problem 65: Right Triangle, Mindpoints.
	Proposed Problem 64: Triangle, Incircle, Transversal.
	Proposed Problem 63: Regular Heptagon, Side and Diagonals.
	Proposed Problem 62: Square Diagonal, Inscribed Circle.
	Proposed Problem 61: Triangle, Trisection of Sides.
	Proposed Problem 60: Isosceles triangle.
	Proposed Problem 59: Right and Equilateral Triangles, Midpoints.
	Proposed Problem 58: Right Triangle, Congruence, Perpendiculars.
	Proposed Problem 57: Angle bisector, circles Cyclic Quadrilateral.
	Proposed Problem 56: Angle bisector, circles Parallel Lines.
	Proposed Problem 55: Angle bisector, circles Cyclic Quadrilateral.
	Proposed Problem 54: Angle bisector, circles Midpoint of arc.
	Proposed Problem 53: Angle bisector, circles
	Proposed Problem 52: Triangle, angles, incircle.
	Proposed Problem 51: Fagnano's Problem Inscribed Triangle with the Minimum Perimeter.
	Problem 50. Triangle with Equilateral triangles.
	Problem 49. Angles and Right triangle.
	Problem 48. Angles and triangles.
	Problem 47. Angles and triangles.
	Problem 46. Angles and triangles.
	Problem 45. Angles and triangles.
	Problem 44. Angles and triangles.
	Problem 43. Angles and triangles.
	Problem 42. Angles and triangles.
	Sangaku Problem 41. "Mickey Mouse." Three circles, tangents, radii.
	Problem 40. Triangle, Incenter, Excenter, Angles 80, 40, Distances.
	Problem 40. Geometry Help. Suggestions.
	Problem 39. Triangle, Incircle, Bisector, Cyclic Quadrilateral and angles.
	Problem 39 Geometry Help. Facts you should know for the proposed problem 39.
	Problem 38. Right Triangle, altitude, incircles, incenters, and angles.
	Problem 37. Right Triangle, altitude, incircles, incenters, and orthocenter.
	Problem 36. Right Triangle, altitude, incircles and inradii.
	Problem 35. Incenters and Inradii in Cyclic Quadrilateral.
	Problem 34. Right Triangle, Cevian, Incircles, Tangents and Inradius.
	Problem 33. Triangle and quadrilateral.
	Problem 32. Triangle, Cevian, Incircles, Tangents.
	Problem 31. Right Triangle, Incircle, Collinears. Problem 30. Right Triangle, Incircle, Inradius.
	Problem 29: Geometry Help. Facts you should know.
	Problem 29. Right Triangle, altitude, incircle and inradius. Ten conclusions.
	Problem 28. Right Triangle, altitude, incircles and inradius.
	Problem 27. Right Triangle, incircles and inradius.
	Problem 26. Right Triangle, altitude, incircles and inradius.
	Problem 25. Right Triangle, altitude, incircles and inradius.
	Problem 24. Right Triangle, altitude, incircles and inradii.
	Problem 23. Right Triangle, altitude, incircles and inradii.
	Problem 22. Right Triangle, altitudes, incircles and inradii.
	Problem 21. Acute triangle, orthocenter, diameter, tangents.
	Problem 20. Right Triangle, incircles and inradii.
	Problem 19. Right Triangle and Excenter.
	Problem 18.
	Problem 17.
	Problem 16.
	Problem 15.
	Problem 14.
	Problem 13.
	Problem 12.
	Problem 11.
	Problem 10.
	Problem 9.
	Problem 8.
	Problem 7.
	Problem 6.
	Problem 5.
	Problem 4.
	Problem 3.
	Problem 2.
	Problem 1.