The mathematician Pascal was most famous for Pascal's triangle, a number pattern that used in the binomial expansion formula. Today, we will introduce a theorem in geometry that bears his name. The Pascal's Hexagrammum Mysticum Theorem states that if we draw a hexagon inscribed in a circle then the three pairs of opposite sides of the hexagon intersect at three points which lie on a straight line.
Ceva's Theorem and Menelaus' Theorem
Today we will learn about two well-known theorems in geometry, Ceva's Theorem and Menelaus' Theorem. These two theorems are very useful in plane geometry because we often use them to prove that a certain number of points lie on a straight line and a certain number of lines intersect at a single point. Both of the theorems will be proved based on a common simple principle. We also generalize the theorems for arbitrary polygons.
Pythagorean Theorem
Pythagorean Theorem is surely one of the most popular theorems in mathematics, which says that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
This theorem has so many proofs. One of the proofs is rather interesting because it was due to James Abram Garfield - the 20th President of the United States.
President Garfield's proof is very simple. It relies on finding the area of the following trapezoid in two different ways.
This trapezoid has two bases of length $a$ and $b$. The height of the trapezoid is $a + b$. So the area of the trapezoid is $$\frac{1}{2}(a+b)(a+b) = \frac{1}{2}(a^2 + b^2 + 2ab).$$
The other way to calculate the area of the trapezoid is to sum up the three areas of the triangles, which is $$\frac{1}{2}ab + \frac{1}{2} c^2 + \frac{1}{2}ab = \frac{1}{2}(c^2 + 2ab).$$
Comparing the two results we obtain the Pythagorean Theorem! $$c^2 = a^2 + b^2.$$
Pythagorean Theorem: $c^2 = a^2 + b^2$. |
This theorem has so many proofs. One of the proofs is rather interesting because it was due to James Abram Garfield - the 20th President of the United States.
President Garfield's proof is very simple. It relies on finding the area of the following trapezoid in two different ways.
This trapezoid has two bases of length $a$ and $b$. The height of the trapezoid is $a + b$. So the area of the trapezoid is $$\frac{1}{2}(a+b)(a+b) = \frac{1}{2}(a^2 + b^2 + 2ab).$$
The other way to calculate the area of the trapezoid is to sum up the three areas of the triangles, which is $$\frac{1}{2}ab + \frac{1}{2} c^2 + \frac{1}{2}ab = \frac{1}{2}(c^2 + 2ab).$$
Comparing the two results we obtain the Pythagorean Theorem! $$c^2 = a^2 + b^2.$$
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