Geometry Explained

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The following Geometry Concepts are explained: Viviani’s Theorem, Proof of Heron’s Formula for the Area of a Triangle, Ratios of special triangles, Finding the Center of a Circle, Radian Measure.

Viviani’s Theorem
Pick a point inside an equilateral triangle and sum its three distances to each of the sides of the triangle. Then this sum has value the height of the triangle, no matter the point initially chosen.
This charmer is attributed to 17th-century mathematician Viviani. Here we prove the result, and then go further and let the point wander outside the triangle, and then go to shapes other than triangles!

Proof of Heron’s Formula for the Area of a Triangle
Around 60 C.E., Heron of Alexandria developed a formula for the area of a triangle in terms of its three side-lengths solely (thus giving no need to find the height of the triangle first). This result is often presented in the school curriculum without proof. Here is its proof!
It is worth noting that the formula was discovered by Chinese and Islamic scholars as well.

On 30-60-90 and 45-90-45 Triangles
A standard geometry course makes kids memorize the ratios of “special’ triangles. Here’s what these triangles really are and how, as a mathematician, one can think your way through them

Finding the Center of a Circle
Here’s a cute challenge from geometry …
Suppose you are given a circle. Use only a piece of paper and a pencil to locate the exact center of that circle!
In this video I show you how to find the exact center of a flower pot, or of the lid of a jar of jam, or of a quarter. It’s fun and represents a neat way to motivate some standard results from the curriculum.

Radian Measure
Mathematicians say that there are 2*π radians in a circle. Allow me to explain what these are (and why they are called “radians”).

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