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Browse High School Triangles and Other Polygons
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Area of an irregular shape.
    Classifying quadrilaterals.
    Heron's formula.
    Polygon diagonals.
    Pythagorean theorem proofs.
    Triangle congruence.

Triangle Area Proofs [01/23/2002]
An analytic proof.

Triangle Centers at Lattice Points [09/03/2002]
Is there a triangle that can be plotted on a rectangular grid so that all of its vertices and all four centers are lattice points? If so, what are the coordinates of the vertices?

Triangle Congruence: AAS and ASA [10/16/2003]
Why isn't AAS (Angle, Angle, Side) used to prove that two triangles are congruent?

Triangle Construction [09/09/2001]
Given a triangle ABC and point D somewhere on the triangle (not a midpoint or vertex), construct a line that bisects the area.

Triangle Construction Given an Angle, the Inradius, and the Semiperimeter [03/26/2002]
Given an angle, alpha, the inradius (r), and the semi-perimeter (s), construct the triangle.

Triangle Construction Given Medians [12/12/2001]
Given median lengths 5, 6, and 7, construct a triangle.

Triangle Construction Given Two Angles and Semiperimeter [03/14/2002]
Given two angles, A and B, and the semiperimeter, construct the triangle.

Triangle Geometry: Sides and Edges [6/2/1996]
If the angles of a triangle are equal, does it necessarily mean that the sides are also equal?

Triangle Inequality Theorem [03/09/2001]
The lengths of the sides of a non-isosceles triangle, in size order, are 5, x, and 15. What are all possible integral values of x?

Triangle: Longest Side Opposite Greatest Angle [10/23/1999]
Prove that in any triangle, the greatest side is opposite the greatest angle.

Triangle Midpoints and Vertices [02/23/2003]
How do you figure out the vertices of a triangle algebraically by using its three midpoints?

Triangle Perimeter [07/20/1997]
How many triangles have sides whose lengths total 15 units?

Triangle Proof [2/18/1995]
Maybe if two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. I'm not sure how to say this in a proof.

Triangle Proof: Parallel Sides and Proportionality [07/27/2004]
How do I prove that a line which cuts two sides of a triangle proportionately is parallel to the third side?

Triangle Proof: r + r1 + r2 = CD [04/20/2001]
Let CD be an altitude of triangle ABC, and assume that angle C = 90 degrees. Let r1 and r2 be the inradii of triangle CAD and triangle CBD, respectively, and show that r+r1+r2=CD, where r is the inradius of triangle ABC.

Triangle Proofs in General [11/19/2001]
Mapping out a general method for proceeding with proofs.

Triangle Proof with Contradiction [02/21/2004]
Let D, E lie internally on side BC of triangle ABC and consider the following conditions: 1) angle BAD = angle DAE = angle EAC 2) |BD| = |DE| = |EC| Prove that, whatever the shape of triangle ABC, 1) and 2) cannot both be true, that is, if either is true, then the other is false.

Triangle Proportions: A Diagram [05/06/2003]
A cone has a circular base radius 1, and vertex of height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What is the length of a side of the cube?

Triangles: Angle Sums [05/15/2002]
Can you draw a triangle in which the sum of any two angles - no matter which two you pick - is always less than 120 degrees?

Triangles in a Polygon [06/14/1997]
A regular 18-sided polygon is inscribed in a circle and triangles are formed by joining any three of the eighteen vertices. How many obtuse triangles are there?

Triangle's Medians Make Smaller Triangles with Equal Area [04/15/1999]
Proving that the six triangles constructed from the three medians of any triangle have the same area.

A Triangle Vertex Bisection and Its Trio of New Lengths [06/08/2012]
A trigonometry student struggles to express where the bisector of a triangle vertex intersects the side opposite it; and to describe the bisector's length in terms of the triangle's side lengths and angle measures. Doctor Peterson unpacks formulas for both, along the way invoking the Law of Cosines — and another doctor's prior work.

Triangle Vertices But Not Sides [02/22/2003]
If P is a regular n-gon, what is the number of triangles whose vertices are the vertices of P but whose sides are NOT the sides of P?

A Triangle with Three Right Angles [12/01/1999]
How can you make a triangle with three right angles?

Triangular Garden [03/18/1997]
Find the length of a fence that runs from the right angle to the hypotenuse and separates the garden into two parts of equal perimeter.

Trisected Hypotenuse of a Triangle [12/20/1998]
In right triangle ABC, with C as the right angle... what is the length of AB (the hypotenuse)?

Trisecting an Angle and the Opposite Side in a Triangle [09/03/2008]
Prove that it is impossible to have a triangle in which the trisectors of an angle also trisect the opposite side.

Truncating a Square to Get an Octagon [10/13/2003]
I want to make an octagon by cutting the corners off of a square. Where do I make the cuts?

Twenty Quadrilaterals from Nine Dots [04/04/1999]
How can you get 20 quadrilaterals from 9 dots?

Two-Column Proof About Kites [11/09/1999]
Can you help me understand a proof about perpendicular lines and congruent triangles in a kite?

Two Column Proof of a Theorem [08/12/1998]
Write a two-column proof and give numbered statements with reasons....

Two Questions on Geometric Harmonics [11/24/2005]
Two circles intersect each other at B and C. Their common tangent touches them at P and Q. A circle is drawn through B and C cutting PQ at L and M. Prove that {PQ:LM} is harmonic.

Two-Sided Polygon? [12/01/2003]
My 5th grade math teacher said that we had to draw a polygon using two straight lines. Is this possible?

Understanding Bearings in Directional Problems [01/14/2004]
A boat sails 10km from a harbor H on a bearing of S30 degree E. It then sails 15 km on a bearing of N20 degree E. How far is the boat from H? What is the bearing from H?

Understanding Rectangle Area and Perimeter [11/08/2002]
True or false: if the perimeter of a rectangle increases, the rectangle's area always also increases.

Uniquely Determining a Polygon [02/05/2001]
Is it true that if you know the side order, side lengths, and area of a polygon, as well as whether each of its angles is obtuse or acute, you have uniquely determined it?

Using Midpoints to Determine Vertices [09/04/2002]
The midpoints of the sides of a triangle have coordinates G(3,1), H (-1,2) and J (1,-3). Determine the coordinates of the vertices of the triangle.

Using the Incenter [05/06/2003]
I need to construct a triangle to fit inside a triangle.

Vectors of Parallelograms and Octagons [07/28/1998]
ABCDEFGH is a regular octagon and AB = p and BC = q. Express AH in terms of p and q...

Venn Diagram to Classify Quadrilaterals [01/02/2003]
I am looking for a Venn diagram that will accurately display the relation among trapezoids, parallelograms, kites, rhombi, rectangles, and squares.

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