EDIT: And because there seems to be a lot of confusion to this (either through people not reading the OP fully or just not understanding the term itself), the key point to the problem is that you can ONLY use elementary geometry to solve this. Nothing beyond what's listed here, so no trig, no algebra beyond the few equations listed, no matrices, etc:
So a sample solution for this type of problem could look like:
Here is everything you need to know to solve the above problems.
Lines and Angles: When two lines intersect, opposite angles are equal and the sum of adjacent angles is 180 degrees. When two parallel lines are intersected by a third line, the corresponding angles of the two intersections are equal.
Triangles: The sum of the interior angles of a triangle is 180 degrees. An isosceles triangle has two equal sides and the two angles opposite those sides are equal. An equilateral triangle has all sides equal and all angles equal. A right triangle has one angle equal to 90 degrees. Two triangles are called similar if they have the same angles (same shape). Two triangles are called congruent if they have the same angles and the same sides (same shape and size).
* Side-Angle-Side (SAS): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.
* Side-Side-Side (SSS): Two triangles are congruent if their corresponding sides are equal.
* Angle-Side-Angle (ASA): Two triangles are congruent if a pair of corresponding angles and the included side are equal.
* Angle-Angle (AA): Two triangles are similar if a pair of corresponding angles are equal.
Here's the actual problem:
EXAMPLE PROBLEM: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
PROOF WRITTEN IN TWO-COLUMN FORM:
PROOF IN PARAGRAPH FORM:
Code:Argument Reason why 1. The two lines marked with one brown little line are congruent. 1. The two diagonals bisect (given). 2. The two lines marked with two brown little lines are congruent. 2. The two diagonals bisect (given). 3. The two angles marked with blue lines are congruent. 3. They are vertical angles. 4. The two yellow triangles are congruent. 4. SAS theorem and 1, 2, and 3. 5. The angles A and A' are congruent. 5. The two yellow triangles are congruent. 6. The angles A' and A'' are congruent. 6. They are vertical angles. 7. The angles A and A'' are congruent. 7. 5 and 6 together. 8. The lines that form bottom and top of the quadrilateral are parallel.8. 7 and the theorem that says that corresponding angles being the same is equivalent to lines being parallel. 9. The lines that form the two sides of the quadrilateral are parallel. 9. Repeat steps 1-8 using the two white triangles. 10. The quadrilateral is a parallelogram. 10. 8 and 9 together.
Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.
Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.
But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.
An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.
So the quadrilateral is a parallelogram.
Using only elementary geometry, determine angle x. Provide a step-by-step proof.
You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc. There is a review of elementary geometry
This is the hardest problem I have ever seen that is, in a sense, easy. It really can be done using only elementary geometry. This is not a trick question.
Here is a very small hint. Here is a small hint.
Tips for Writing Proofs
Proofs may be written informally using plain English. Just be sure to include all the steps in your reasoning, or at least all the key steps. Providing a diagram is very helpful but not required. You can draw a diagram on the computer or you can draw it on paper and then scan it or photograph it with a digital camera. Name each point you use with a letter. If you don't provide a diagram, you will need to describe the named points with words (ex., say "the intersection of AE and DB is G "). Identify lines with two letters (ex., say "line AB" or simply "AB"). Identify triangles with three letters (ex., say "triangle ABC" or "tri ABC" or simply "ABC"). Identify angles with three letters, vertex in the middle (ex., say "angle ABC" or "ang ABC" or "<ABC" or simply "ABC"). It is helpful to number your steps to make it easy to refer back to earlier steps.
Please don't search the the web for the answer -- that's cheating. You will only deprive yourself of many hours of delicious frustration.
I did not invent these problems. After I first read problem 1, I worked on it for many hours over several days before I eventually figured it out. A couple of years later I came back to the problem, but I had forgotten my proof. It took me many hours to figure it out again! Problem 2 also took me many hours to solve.
How hard are these problems? Any teenage student and some younger students can understand the proof, but very very few are able to discover the proof on their own. Of the hundreds of people that have emailed me, I'd estimate only one or two percent (mostly math professionals and college students) have solved it without significant hints. (The hints given above are not significant hints.) Most people who think they have found the solution are wrong.
These problems have been published in many places. Problem 2 first appeared here: Langley, "A Problem", Mathematical Gazette, 1922. Dr. Gary Gruber says his high school teacher showed him problem 1 in about 1955. Tom Rike says problem 1 first appeared in print here: Harry Schor, The New York State Mathematics Teachers' Journal, 1974. It also appeared here: "Problem 134", Eureka (now Crux Mathematicorum), 1976. Dr. Gruber popularized problem 1 in several papers (such as "The Genius Test") which appeared in newspapers throughout the 1990s (Universal Press Syndicate and Los Angeles Times Syndicate). That's where I discovered it.