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Browse College Triangles and Other Polygons

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Area of Equilateral Triangle [12/06/2002]
Equilateral triangle ABC has, near its center, point P, which is 3, 4, and 5 units from the three vertices. What is the area of triangle ABC?

Area of Triangles When Altitudes Are Given [12/16/2002]
Find the area of a triangle when the lengths of the three altitudes are given.

Building Squares with Tangram Puzzle Pieces [03/21/2006]
Is it possible to build a square using only 6 of the 7 pieces of the well-known Tangram puzzle?

Calculating Polygon Area [01/11/2004]
How can you determine the area of an unusually shaped polygon?

Classical Geometry [04/16/2002]
Let ABC be a triangle with sides a, b, c. Let h be the perpendicular from A to a, and m the median from A to the midpoint of a. Construct the triangle using only ruler and compass if you know A, h, m.

Cross-Cornering a Shape to Make it Square [12/19/2002]
We lay out a building that is 30' x 40' and cross-corner it to see if it is 'square,' but there is a 6' difference. What is the equation to find how far to move one side to make the shape 'square'?

Curious Property of a Regular Heptagon [04/06/2001]
How can I prove that in a regular heptagon ABCDEFG, (1/AB)=(1/AC)+(1/ AD)?

Derivation of Law of Sines and Cosines [11/02/1997]
How do you derive the law of sines and the law of cosines?

Determining If a Given Point Lies inside a Polygon [12/27/2005]
I have a finite number of points that constitute a polygon, and a point p(x,y). I want to know if point p lies inside the polygon.

Drawing Regular N-gons (Compass and Straightedge) [11/17/1997]
Is it true that the only regular n-gons that can be drawn using ONLY a straightedge and compass are those with the number of sides equal to a Fermat Prime or a product of Fermat Primes?

Equal Parallelians Point [12/09/2002]
Within triangle ABC, draw three segments parallel to the sides of the triangle, each touching two sides. The three segments meet at one point, and they are all the same length, x. Find the length of x given the length of the sides of triangle ABC.

The Erdos-Mordell Theorem [10/13/2000]
Let P be a point in a triangle, D be the sum of the distances from P to the 3 vertices, and E be the sum of the distances from P to the edges. How can I prove that D is greater than or equal to 2*E?

Euler in the Product of a Regular Polygon's Diagonal Lengths [04/06/2010]
A professor emeritus considers an n-sided regular polygon A1, A2, ... An inscribed in the unit circle; and conjectures that the product of the lengths of its diagonals equals n. By defining the polynomial f(x) as the product of x - r over its (n - 1) roots, and applying complex numbers and Euler's equation, Doctor Vogler proves that sin(pi/n) * sin(2pi/n) * ... * sin[(n - 1)pi/n] = n/2^(n - 1).

Euler's Line Theorem [04/08/2001]
Prove that the circumcenter, orthocenter, and centroid of any triangle lie on the same line using analytical geometry.

Existence of the Brocard Point [06/06/2002]
Demonstrate that the Brocard point exists in any triangle.

Explanation and Informal Proof of Pick's Theorem [04/27/2004]
We just learned Pick's Theorem, A = b/2 + I - 1, where b is the boundary pegs, I is the interior pegs, and A is the area. I don't get why it works. Why do you divide the boundary by 2 and subtract 1?

Explanation and Test Case for Pick's Theorem [06/13/2006]
I don't really understand Pick's Theorem and its formula. Can you explain the formula and show how it works for a polygon?

Finding Side Lengths of a Scalene Triangle [6/2/1996]
Two observers on points A and B of a national park see a beginning fire on point C. Knowing that the angles CAB = 45 degrees, ABC = 105 degrees and that the distance between points A and B is of 15 kilometers, determine the distances between B and C, and between A and C.

Finding the Area of an Irregular Polygon [02/23/2008]
What is the formula for finding the area of an irregular polygon?

Finding the Base of Parts of a Triangle [05/22/2000]
Can you derive an expression for L1 in terms of L2 and L3 such that the area of a triangle with base A1 and the area of a triangle with base A2 are each 10% of the total area?

Find the Area of the Quadrilateral [05/20/2003]
Find the area of an irregular quadrilateral formed by the given intersection of two squares.

Geodesics [12/15/1996]
Can you give me information on the math behind geodesics?

Geometry Proof Involving Circle and Triangle [09/26/2005]
Triangle ABC cuts a circle at points E, E', D, D', F amd F'. Prove that if AD, BF and CE are concurrent, than AD', BF' and CE' are also concurrent.

Incenter and Conway's Circle [12/17/2002]
In a triangle, the bisectors of the angles intersect at a point in the interior of the circle. If I use this point as a center to draw a circle, what is the relation of this circle to the triangle?

Incenters, Orthocenters, and the Spieker Point [02/13/2000]
Prove that the circumcenter of a triangle is the orthocenter of its medial triangle, and that the incenter of the triangle is the orthocenter of the triangle formed by the 3 excenters.

Inclusive Definitions: Trapezoids [11/04/2004]
As far as I know, a trapezoid is defined as a quadrilateral with exactly one set of parallel sides. However, a very highly regarded educator and textbook author recently argued that this definition is incorrect. His definition of a trapezoid is that it is a quadrilateral that has at least one pair of parallel sides. A square, therefore, would be considered a trapezoid. Is he correct or are thousands of books going to be published with the wrong definition?

Inconstructible Regular Polygon [02/22/2002]
I've been trying to find a proof that a regular polygon with n sides is inconstructible if n is not a Fermat prime number.

Intersections of Polygon Diagonals [8/26/1996]
Given a regular polygon, write a formula for the number of zones it is divided into by its diagonals.

Isosceles Triangle Maximizes Area? [09/11/2003]
How can you show that among all triangles having a specified base and a specified perimeter, the isosceles triangle on that base has the largest area?

Magic Triangle Puzzle [07/26/2002]
Where did the white square come from?

Maximizing Irregular Polygon Area: Which Circle? [05/01/2011]
How do you determine the radius of the circle that maximizes the area of an irregular n-gon circumscribed on it? With the Pari computer algebra system, Doctor Vogler approaches the question using numerical techniques such as Newton's Method and a binary search, which suggests that no closed-form expression exists.

Maximum Number of Acute Angles in a 2001-gon [05/29/2002]
What is the largest possible number of acute angles a 2001-gon can have if no two sides cross each other?

Maximum Quadrilateral Area [05/15/2001]
Given a quadrilateral with sides of lengths a,b,c,d, prove that its area is maximized when opposite angles are supplementary.

Maximum Rectangle within a Quadrilateral [10/25/2001]
I need to extract from a quadrilateral the maximum area rectangle inside it.

Maximum Surface Area [07/03/2003]
Within a rectangle x by y, I wish to draw a shape that is no more than x across in any direction, but which has the largest possible surface area within the confines of the rectangle.

Measures of Interior and Exterior Angles of Polygons [07/07/2005]
A question about star polygons leads to a discussion about calculating interior and exterior angles of polygons.

Miquel Circles [09/17/2003]
Given an acute triangle ABC, consider all equilateral triangles XYZ, where points A, B and C lie on segments XZ, XY, YZ. Prove that all centers of gravity of all these triangles XYZ lie on one circle.

Nonagon or Enneagon? [02/06/2003]
Is 'enneagon' really the correct name for a 9-sided polygon?

Number of Regions in a Convex Polygon [04/29/2008]
How many regions is a convex polygon divided into by its diagonals, if no three diagonals are concurrent (intersect at a single point)?

Overlapping right triangle problem [09/14/1997]
Given right triangles ABC and DCB with rt angles at B and C, triangle ABC's hypotenuse 20 and triangle DCB's hypotenuse 30. The hypotenuses intersect at point E, a distance of 10 from BC. Find the length of BC.

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