Geometry

This is the Geometry page. I will add concepts on request by email or comments. I don't have a whole lot of time to post, so I will do my best to post when I can.





The Importance of Symbols-
To convince you that names and symbols are useful, we'll start at the end of the book instead of the beginning. Here's the final example problem in this book, written without any special symbols or names.
Draw three points and connect each to the other two with straight paths. Also, draw the circle that passes through all three of these points. Then, draw a line through one of those three points such that the line goes inside the region that you just formed and is equally close to the other straight paths you formed initially through this point. Draw the circle that goes through the one of your three first points you just drew a line through, through the point where this line hits the straight path that connects the other two of your first three points, and through the point that is half-way between these two other points.
Consider the two paths from the point we drew the extra line through to the other two of our first three points. These paths hit our second circle before they hit these other two points. Show that the distance from where the circle hits these paths to the points where these paths end is the same for both paths.
Quote from "Introduction to Geometry" from Art of Problem Solving, by Richard Rusczyk




Sorry. Skipping ahead because I wasn't able to post for a while, and I can't remember what happened during that time, so please forgive me.


Pythagorean Theorem -
The Pythagorean Theorem was most likely introduced in eighth grade math or Algebra 1. If you have already done those, you might remember this, but for those who don't, here it is.

Basically, the Pythagorean Theorem states that for all right triangles, the value of the length of the two shorter sides (the legs) squared (individually; not added then squared) then added together equals the length of the hypotenuse. The equation is:

  

A and B are the legs, and C is the hypotenuse (for those who don't know, the hypotenuse is the leg that is the longest in a triangle, and in a right triangle, it is the side opposite the right angle). There are many proofs for this, so I won't show many (will put them in later).

Heron's Formula -

Heron's Formula is introduced in Geometry. It allows you to find the area of any kind of triangle, so long as you have all the side lengths. To use Heron's Formula, you need the length of all the sides of the triangle.

To find the area of the triangle, use the side lengths to find the perimeter and then get the semiperimeter (for those who don't know what that means, that is half of the perimeter). Then use this equation:

S is the semiperimeter, and a, b, and c are the triangle side lengths. The steps are: 1. find the semiperimeter; 2. write the equation shown above; 3. do prime factorization for all the numbers; 4. simplify by finding all pairs, then crossing out one of the numbers in the pairs, then leaving the numbers without a pair under the square root. Example:

This formula works for any kind of triangle, no matter what the side length is (I'm not sure about negatives, so I will check and then post the answer on the main page (it will be a yes or no)).

Angle Bisector Theorem -
The angle bisector theorem is a helpful theorem for finding side lengths in a triangle if you are given two or three side lengths. If you don't have the side lengths, then that might be a problem.