Prove It To Me

The proof portfolio at the end of the semester needs at least one proof from each set. Presentation of the proof should be very neat. The ideal is that someone not working on that problem could easily follow your proof. Please use, like, you know, good English. (Proper Spanish also accepted.)
 

Set One
one proof due by 10/6/99

1) Four Line Geometry. There exists a pair of points in the geometry not joined by a line.

 2) Fano's Geometry. Thelines through any one point of the geometry contain all the points of the geometry.

 3) If a motion has three noncolinear fixed points, the motion is the identity. (You may use the following theorem: A motion of the plane is uniquely determined by an isometry of one triangle onto another.)

 4) The image of the parabola y=ax2 under a dilation with fixed point (0,0) is a parabola. 

Set Two
one proof due by 10/20/99

1) Transformations. The angle between two intersecting lines in the plane is invariant under any reflection.

 2) The segment joining the midpoint of two sides of a triangle is parallel to the third side.

 3) A triangle and its medial triangle (triangle with the midpoints of the first triangle's sides for vertices) have the same centroid.
(cf. p.8, Active Geometry)

 4) The radius of the circumcircle of a triangle (the circle through the three vertices) is twice the radius of the ninepoint circle for the same triangle.
(cf. p.13, Active Geometry) 

Set Three
one proof due by 11/10/99

1) If the three points on the sides of the triangle are collinear, the Miquel point is on the circumcircle. 
(cf. p.17, Active Geometry)

 2) If the Miquel point is on the circumcircle, then the three points on the sides of the triangle are collinear. 
(cf. p.17, Active Geometry)

 3) The altitudes of a triangle are the isogonal conjugates of the circumradii to the vertices.
(A picture is here as THM3-3.GSP in sketchpad format. Click on the file name with right mouse button and choose "Save Link as...")

 4)Let S be the symmedian point of a given triangle and D, E, and F the feet of the perpendicular segments from S to the sides of the triangle. S is the centroid of triangle DEF. 
(A picture is here as THM3-4.GSP in sketchpad format. Click on the file name with right mouse button and choose "Save Link as...") 

Set Four
TWO constructions due by 11/29/99

Include a step by step description of the construction. May be done by hand or on sketchpad (using only circle with radius and straight edge).

 1) Given a ratio a/b (i.e. given two lengths a and b) construct four collinear points A, B, C, and D such that AB/BC=AC/CD=a/b.

 2) Construct a triangle given the lengths of one side, an altitude to a second side and the circumradius.

 3) Construct a circle tangent to two given intersecting lines and a radius.

 4) Construct a parralelogram given the length of a diagonal, the length of one side and a length numerically equal to the area. 

Set Five (aka Last)
one proof due by Wednesday of Finals week, 11/15.

1) In four line geometry, there exists exactly six points. 

2) Given an equilateral triangle and a point in the triangle, the sum of the lengths of the perpendiculars from the point to the sides is equal to the length of the altitude.

3) Construct a triangle given the lengths of two medians and the length of the altitude to the third side. 

4) Any of: One-3, Two-4, or Three-4 from previous sets. 

Final Proof Portfolio:  12/15, Wednesday of Finals Week

1 proof from each set (does not have to be the original submission), graded for 10 points each.

At this point, correctness is assumed.  Five points will be removed for an incorrect proof, 3 points for a gap of consequence.

The grading will be mostly on clarity of presentation.  Consider the use of colors, explanatory pictures and complete sentences.  Avoid abbreviations, citing of unstated theorems and pages of equations.  A proof should read well -- have one of your friends not in this class be a proof reader. (Origin of the expression?)  Is your proof understandable to a mathematically competent person who has yet to think about the particular problem?  Ways to help this:  state what you are going to prove and how you are going to do it.  Define important terms which are not common. (Eg. isogonal.)  Make it beautiful!
 
 

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

Edna St. Vincent Millay, (1892 - 1950)