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Dr. Math FAQ:
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.
Pythagorean theorem proofs.
- Maximum Area of Inscribed Triangle [12/10/2001]
An isosceles triangle is inscribed in a circle of radius R. Find the
value of Theta that maximizes the area of the triangle.
- 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 Rectangle within a Quadrilateral [10/25/2001]
I need to extract from a quadrilateral the maximum area rectangle inside
- 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.
- Measuring by Shadows [05/22/2001]
How can I measure a tree using its shadow and mine?
- Measuring the Height of a Building Using Shadows [05/24/2000]
What time of day is best to use a shadow to measure the height of a
building by using triangles?
- Median and Altitude Constructions [07/16/2003]
How to draw a median and an altitude from the three sides of a
- Medians of a Triangle [02/16/1999]
Prove that the 3 medians of a triangle divide themselves up into a ratio
- Medians of Triangles Proof [05/29/2000]
Prove that in any triangle, the sum of the medians is more than 3/4 of
its perimeter, but less than the whole perimeter.
- Menelaus's Theorem [01/25/1999]
A straight line intersects sides AB, BC and the extension of side AC of a
triangle ABC at points D, E and F respectively. Prove that the midpoints
of the line segments DC, AE and BF lies on a straight line.
- Mid-segment Theorem [02/02/1999]
Can you help me prove the Mid-Segment Theorem?
- Minimal Distances to a Point in a Triangle [01/05/2001]
How can I prove that the smallest value of PA + PB + PC occurs when angle
APB = angle BPC = angle CPA = 120 degrees, for a triangle ABC and a point
- Minimum Angle Proof [07/05/2001]
Label the point of intersection of the angle bisectors of triangle ABC as
Q. Let M be the midpoint of side BC. Given that MQ = QA, find the minimum
value of angle MQA.
- 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.
- Mixtilinear Incircle Proof [12/11/2000]
In triangle ABC, AB = AC. A circle is tangent internally to the
circumcircle of triangle ABC and also to sides AB and AC at points P and
Q, respectively. How can I prove that the midpoint of the segment PQ is
the center of the mixtilinear incircle of triangle ABC?
- Nagel Point [04/15/2001]
What relation does the Nagel Point have to the incenter, centroid, and
Spieker point of a triangle?
- Naming Corresponding Parts of Congruent Figures [04/23/2003]
Given triangle OPS congruent to triangle TQR, name the corresponding
sides and angles of the two triangles.
- Naming the Isosceles Triangle [09/23/1998]
How did the isosceles triangle receive its name?
- The Napoleon Point and More [09/04/1998]
How do you prove that the Napoleon point will always exist, given the
proper conditions? Is there a stronger theorem?
- Napoleon's Triangle [08/10/1999]
What is Napoleon's triangle?
- Nine-Sided Polygon [06/11/1997]
Can you construct a regular 9-sided polygon inside a circle using only a
compass and straight-edge?
- Nonagon or Enneagon? [02/06/2003]
Is 'enneagon' really the correct name for a 9-sided polygon?
- Non-Congruent Triangles [12/12/2001]
Construct and prove that there can be two non-congruent triangles in
which five parts of one triangle are equal to five parts of another.
- Nonconvex Polygon Angle Measure [02/03/1999]
What is the formula to find the interior angle measurements of a
- Number of Lines of Symmetry in a Regular Polygon [03/12/1998]
In a regular polygon, is the number of lines of symmetry the same as the
number of lines or angles of that polygon?
- Number of Points in a Star [7/16/1996]
Is there a way to predict the number of points in a star given only the
internal angle of the corners?
- Octagon Construction Using Compass Only [02/22/2002]
Construct the vertices of a regular octagon using just a compass. The
only thing you know about the octagon is the circumradius.
- Octagon Formula [07/30/1997]
If you're building an octagon on a 12-foot radius, what is the length of
- Octagon Side Lengths [08/22/2001]
If I know that the dimension of an octagon from one side to the other is
8 feet, how can I find the lengths of a side?
- One- and Two-sided Polygons [12/07/1999]
Can you explain what a monogon and a digon are?
- Order of a 3D Triangle [08/29/2001]
If I visit the vertices of a 3D triangle in order going from a to b to c,
am I going clockwise or anticlockwise?
- Origami Equilateral Triangle [04/26/2001]
How can I create an equilateral triangle from a piece of paper using only
- Orthic Triangle [04/09/1999]
How do you find the angles of the triangle ABC, similar to triangle
A1B1C1, where AA1, BB1, and CC1 are the altitudes of triangle ABC?
- 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.
- Parallelogram Perimeter [10/01/1997]
The diagonals of a parallelogram are 10 and 24 in length. If one side of
the paralellogram is 13, what is the perimeter?
- Parallelogram Side Length [08/12/2003]
In a parallelogram ABCD, K is a point on AB such that angle DKC = 90
degrees and AD = AK. If AD = 10cm, find the length of AB.
- Pascal's Theorem [10/07/1998]
Can you explain Pascal's Theorem? How does it relate to conic sections?
- Pedan Trapezium [05/23/2000]
How can I prove that a isosceles trapezium whose parallel side lengths
are 7 and 4 respectively, and whose slant sides have length 6, is a Pedan
- Pentagon Area Using No Trig [05/14/2001]
Where I am stumped is finding the area of one of the five triangles.