1. Begin by writing the formula presented in this unit for finding the area of a triangle, using angle C. 2. Using the Pythagorean identity, sin2C + cos2C = 1, solve for sinC.
Note: Use only the positive root.

3. Substitute the expression from step two into the equation in step one.
4. Using the law of cosines (c2 = a2 + b2 - 2abcosC), solve for cosC. (Rewrite the equation so that you have cosC isolated on one side of the equation.)
5. Square both sides of the equation in step four to get an expression for cos2C. Do not expand the numerator of this fraction! Leave it written as a quantity squared.
6. Substitute this expression for cos2C into the equation in step three. Get a common denominator and simplify enough to eliminate the fraction. Do not expand and simplify the numerator!
7. Checkpoint! At this time you should have the following expression:
K = 1/4 (sqr root)4a^2-(a^2 + b^2 + c^2)^2

If you do not have this equation, you may wish to try to find the error in your work above.

8. Using the format for factoring the difference of two squares [a2 - b2 = (a - b)(a + b)], factor the expression under the radical given in step seven. Do not expand quantities.
9. The result for this step in the proof is provided for you below. Your task is to show that this is in fact the result of manipulating the expression in step eight.
Hint: Remove the parentheses only and regroup.
Step nine: 1/4 (sqr root)[c^2 - (a - b)^2][(a + b)^2 - c^2]

10. Using the format for factoring the difference of two squares [a2 - b2 = (a - b)(a + b)], factor the expression under the radical in step nine.
11. You're almost done! The last step is to convert this expression to Heron's formula. You will need to rely on the definition of s to do this step. Here is some help to get you started:
Recall that s = (a + b + c).

Multiply both sides by 2:
2s = a + b + c.

Now, subtract 2c from both sides of this equation:
2s - 2c = a + b + c - 2c
2s - 2c = a + b - c

Note also that 2s - 2c = 2(s - c).

Now use algebra to make the substitutions and show that the expression in step ten is equal to Heron's formula.

12. The abbreviation Q. E.D. is often written at the end of a mathematical proof. Look up its origin and meaning, and write it in the box below. If appropriate, write it at the end of your proof.